question about length contraction

[gib65]

A friend asked an interesting question about length contraction that I didn't know the answer to: as we know, the faster an object travels, the more it's length contracts. But what happens when you have two objects travelling, one behind the other, at the same rate at the same time? For example, two trains on a track, one behind the other, and they both start moving at the same rate at the same time. What is the center point around which length contraction occurs. Does each train contract around its own center, or can both trains be considered one object and their collective length contracts around the mid-point between them?

Another way of asking this is: what counts as an "object" in relativity?

[swansont]

 

A friend asked an interesting question about length contraction that I didn't know the answer to: as we know, the faster an object travels, the more it's length contracts. But what happens when you have two objects travelling, one behind the other, at the same rate at the same time? For example, two trains on a track, one behind the other, and they both start moving at the same rate at the same time. What is the center point around which length contraction occurs. Does each train contract around its own center, or can both trains be considered one object and their collective length contracts around the mid-point between them?

Another way of asking this is: what counts as an "object" in relativity?

 

 

Relative to each other, nothing is happening. For an observer in another frame, the length contracts, which would include the length of space between them. if the trains were 100m long and 100m between them, and they moved such that gamma=2, then each train would be 50m and there would be 50m between them.

 

Length contracts. Not objects.

[xyzt]

 

A friend asked an interesting question about length contraction that I didn't know the answer to: as we know, the faster an object travels, the more it's length contracts. But what happens when you have two objects travelling, one behind the other, at the same rate at the same time? For example, two trains on a track, one behind the other, and they both start moving at the same rate at the same time. What is the center point around which length contraction occurs. Does each train contract around its own center, or can both trains be considered one object and their collective length contracts around the mid-point between them?

Another way of asking this is: what counts as an "object" in relativity?

 

To put swansont's post into math:

 

An object of length [math]L[/math] is located at distance [math]d[/math] from its frame of reference. So , its extent is [math][d, d+L][/math].

 

In another frame, moving at speed [math]v[/math] wrt the object, its extent is given by the Lorentz transforms, so the extent is:

 

[math][d \sqrt{1-(v/c)^2}, (d+L) \sqrt{1-(v/c)^2}][/math].

 

So, the object distance to the origin of the system of coordinates contracts just as its length.

 

A train can be described as:

 

[math][(d_1, d_1+L_1),(d_2, d_2+L_2)][/math] in its co-moving frame. In the moving frame, the train is described as:

 

[math][(d_1, d_1+L_1)\sqrt{1-(v/c)^2},(d_2, d_2+L_2)\sqrt{1-(v/c)^2}][/math]

Edited January 27, 2015 by xyzt

[md65536]

 

they both start moving at the same rate at the same time.

 

Careful because there is no universal meaning of "at the same time". It will be different for someone beside the tracks compared to someone on the moving trains. For example...

 

 

if the trains were 100m long and 100m between them, and they moved such that gamma=2, then each train would be 50m and there would be 50m between them.

 

This is true for an observer at rest relative to the tracks, if the trains are 100 m long with 100 m between in the moving trains' frame.

 

If all parts of the trains start at the same time according to an observer on the tracks, they'll remain 100 m long with 100 m between them in the track's frame.

 

 

Is this helpful or confusing? Unfortunately you can't investigate all the details of a thought experiment like this one without being precise about whose frame measurements are specified in.

 

The question can be simplified by considering only inertial frames and ignoring when the trains start moving. If the trains are moving with gamma=2 and are 100 m long with 100 m in between them in their own frame, then they'll be 50 m long and 50 m between them in the track's frame. The answer is that all lengths in the direction of motion are contracted.

 

(Just for complication, how would a train have to begin moving in order to get this situation? You can imagine the trains starting at rest in the track frame, and an inertial "ghost train" that eventually lines up with the moving trains. From the track the ghost trains are 50 m long and separated by 50 m, so the end of the last train will have to start moving first to line up with the ghost train, then the start of the second train, etc until finally the start of the forward train begins moving last. Meanwhile from the perspective of the moving trains's frame, the ghost trains are 100 m with 100 m between, while the trains at rest relative to the track are 50 m long with 50 m between. From this frame, the front of the forward train must begin moving first, and the end of the trailing train must begin moving last. Fun!

 

Also: It would be a frame in which the tracks and the moving train have the same speed but in opposite directions, in which the trains started moving at the same time. They're length contracted the same whether at rest relative to the tracks, or traveling in the opposite direction as the tracks.)

Edited January 28, 2015 by md65536

[gib65]

Thanks everyone for your answers.

 

I guess I'm still confused. I'm just wonder why "the middle" should be considered to be exactly mid-way between the two trains. I mean, why not somewhere around Jupiter? (assuming Jupiter is in the direction of their travel). Why doesn't all length contract around a "center" which is placed light-years away from Earth, in which case the trains, once they get up to speed, will be positioned way out in space?

 

I think I *might* know the answer to this, but you tell me: all events that happen for one observer must happen for all observers--so if a passenger on one of the trains experiences trains begin at some point on the track, pass by a station, and stop at another station, those events must happen for everyone (including observers at the stations). So the question isn't where length contraction ends up positioning the trains, but when each event happens. This includes when each point on each train passes by each point on the track (say, for example, somewhere between the first station and the last station). That the trains seem to be contracted from the point of view of someone at the first station just means that the time at which the front of the train passes by a point on the track close to the station will be judged to be closer in time to that when the rear of the train passes by that same point (closer in time, that is, compared to someone on the train). If these two events--the front of the train and the rear passing by a certain point--occur closer in time than they would according to a Newtonian perspective, then one can only conclude that the length of the train must also be less than it would according to a Newtonian perspective.

 

In other words, it's a matter of there being no pre-determined time at which certain points on the trains must pass by certain points on the track (no pre-determined time, that is, on which all observers will agree), and length contraction is simply a consequence of this. Am I in the ball park?

[swansont]

I guess I'm still confused. I'm just wonder why "the middle" should be considered to be exactly mid-way between the two trains.

Because that's how "middle" is defined?

[xyzt]

Thanks everyone for your answers.

I guess I'm still confused. I'm just wonder why "the middle" should be considered to be exactly mid-way between the two trains.

If you look at the math in my answer to you , you will realize that "middle" is a frame-dependent notion, it means one position in the frame of the train and a DIFFERENT position in the frame of the track.

 

 

 

If these two events--the front of the train and the rear passing by a certain point--occur closer in time than they would according to a Newtonian perspective, then one can only conclude that the length of the train must also be less than it would according to a Newtonian perspective.

 

Correct, this is a manifestation of length contraction. It is responsible for us being able to "cram" more particles in a particle accelerator than predicted (and allowed) by Newtonian mechanics. Special Relativity trumps Newtonian mechanics.

Edited January 28, 2015 by xyzt

[swansont]

In other words, it's a matter of there being no pre-determined time at which certain points on the trains must pass by certain points on the track (no pre-determined time, that is, on which all observers will agree), and length contraction is simply a consequence of this. Am I in the ball park?

I think so. Events occurring are absolute (if something happens, it happens in all frames), but details of length and timing are relative to the frame in which they are measured.

[md65536]

Why doesn't all length contract around a "center" [...]

 

[...]

 

Am I in the ball park?

 

It sounds like you understand it.

 

There's no "center of contraction" similar to how there's no center of the universe yet it is expanding in all directions. The contraction (expansion) is uniform... the same at any point. Yes, I would say that picking a center is similar to picking the time that an arbitrary point on an infinitely long train passes an arbitrary point on infinitely long tracks. You could choose different points and it happens at a different time.

 

In other words, it's a matter of there being no pre-determined time at which certain points on the trains must pass by certain points on the track (no pre-determined time, that is, on which all observers will agree), and length contraction is simply a consequence of this.

 

You could as easily say that relative simultaneity is a consequence of length contraction and time dilation, or that all three are a consequence of the constant speed of light (or vice versa). They all fit together, no matter which one you start with.

[phyti]

1. The objects are independent of each other.

2. If they moved at different speeds, that would require space to contract at two different rates simultaneously!

3. There is no known substance to space that can contract.

4. The contraction involves em fields.

5. The solution to the Bell spaceship problem is decided on the basis of the gap increasing as the ships contract, which stresses the connecting "rope".

6. The separation of the objects contracting along with the objects would be perceived by a viewer moving past them at high speed.

7. In the launch frame U, assume the objects contract from the front, while the back ends maintain a constant separation. The gap increases. To maintain a constant gap, the rear object would have to increase speed, or the front object would have to decrease speed. I.e. they could not follow the same speed profile.

[Strange]

2. If they moved at different speeds, that would require space to contract at two different rates simultaneously!

 

In that case, the objects are moving apart and so the space between them will be changing for that reason.

 

3. There is no known substance to space that can contract.

 

There doesn't need to be: we are just talking about the distance that is measured. You can measure distance in space without a substance present.

 

4. The contraction involves em fields.

 

Don't be rdiculous.

[JohnSSM]

 

 

Don't be rdiculous.

Ha...

 

Em fields? Explain please.

If youre talking about 2 trains moving at the exact same speed, in the same exact direction, you could actually just consider it one long train with a gap...

[phyti]

The hyperbola is a constant time t=1, wherever it intersects the path of a moving object. In the A frame (black) light requires 1 unit of time to reflect from the end of a stick .5 units long. If the same measurement is expected in the B frame (red), the same stick requires a physical length contraction.

The space between 2 sticks in tandem would increase, not contract. The volume of space remains unchanged.

Edited February 5, 2015 by phyti

[whiskers]

It's important to remember that the people in the trains could say that they are not moving at all and they are not wrong. They find no contraction of anything which is not moving as far as they can tell.

[phyti]

 

Don't be rdiculous.

wikipedia

tests of special relativity

time dilation and length contraction

Direct confirmation of length contraction is hard to achieve in practice since the dimensions of the observed particles are vanishingly small. However, there are indirect confirmations; for example, the behavior of colliding heavy ions can only be explained if their increased density due to Lorentz contraction is considered. Contraction also leads to an increase of the intensity of the Coulomb field perpendicular to the direction of motion, whose effects already have been observed. Consequently, both time dilation and length contraction must be considered when conducting experiments in particle accelerators.

Don't be ignorant.

[Strange]

 

wikipedia

tests of special relativity

time dilation and length contraction

Direct confirmation of length contraction is hard to achieve in practice since the dimensions of the observed particles are vanishingly small. However, there are indirect confirmations; for example, the behavior of colliding heavy ions can only be explained if their increased density due to Lorentz contraction is considered. Contraction also leads to an increase of the intensity of the Coulomb field perpendicular to the direction of motion, whose effects already have been observed. Consequently, both time dilation and length contraction must be considered when conducting experiments in particle accelerators.

Don't be ignorant.

 

 

That has nothing to do with what you claimed, that length contraction is due to EM fields.

[gib65]

Here's something that confuses me about length contraction:

 

Suppose you had a toy train Tn1 on a straight track at position P1 (at Tn1's center). You turn on the voltage and watch Tn1 accelerate. It accelerates at rate A1 for an amount of time T1, covering a distance D1, and arriving at position P2 (again, at its center). (Note that Tn1 doesn't stop or slow down on or before P2, it just arrives there).

 

Now you remove Tn1 and place another train Tn2 on the track, only this time you position Tn2 at position P3 which is 20 meters ahead of P1 (where D1 > 20 meters). Again, you switch the voltage on. Tn2 accelerates at A2 for time T2, covering a distance D2, after which it arrives at position P4.

 

Now, given that the conditions of the track do not change throughout this experiment and that Tn1 is identical to Tn2, it would be fair to say that A1 = A2 and that if T1 = T2, then D1 = D2 and P2 - P1 = P4 - P3.

 

So far, none of this denies that Tn1 and Tn2 length contracted as they accelerated. We can say that Tn1 contracted by an amount L1 and that Tn2 contracted by an amount L2, and that L1 = L2.

 

But now, you conduct a third experiment. You repeat the previous two experiments in tandem: both Tn1 and Tn2 are placed on the same track at positions P1 and P3 (respectively). The track is turned on. One would expect both trains to accelerate at A1=A2 and if allowed to continue for time T1=T2, then they should each individually cover distance D1=D2 to arrive at position P2 and P4 (respectively). In other words, nothing from the previous two experiments should change just because we allow the two trains to accelerate at the same time.

 

I would expect both to still contract by L1=L2, but the part that has me confused (still) is that the space between them will also contract. What this seems to mean to me (from my most likely inadequate understanding) is that, from the point of view of a person standing still watching the trains go by, the Tn1 will look like it is "catching up" to Tn2--either that or Tn2 is "slowing down" (or at least not accelerating as fast) compared to Tn1--they are closing the gap between them, in other words.

 

This is what it would look like, at least, from the point of view of someone standing still watching the trains go by.

 

So, that person would have to measure that A1 > A2.

 

Maybe that's just the case. Maybe that's just what happens according to relativity. But it seems really odd that simply adding an extra train to the track would change the results of the experiment. Something's not right here.

Edited February 13, 2015 by gib65

[imatfaal]

Just to be clear why do you assert that the gap changes? The "space" is not moving and neither is the track (from the frame of the observer). The gap gets smaller from the frame of reference of the train - as distance of the "moving track" is contracted from the frame of the train in which the train is stationary. The train sees the track contracted, the observer sees the train contracted - neither sees both contracted.

 

I haven't really analysed what a stationary observer will see - you will have to be very careful with simultaneity - but I think we could progress better ifyou explain why the gap closes from the perspective of the trackside obverser

[md65536]

 

I would expect both to still contract by L1=L2, but the part that has me confused (still) is that the space between them will also contract.

If you force various points to remain synchronized in the track frame, the distances between those points will remain the same in the track frame. Effectively you'll be stretching the distance between points (the gap will be increased in the moving train's rest frame) to exactly counteract length contraction. This is set up like Bell's paradox. The answers to your questions can probably be found in an explanation of the paradox.

 

If you don't want to deal with the details of relativity of simultaneity, just give the train a fixed rest length, and let it remain moving inertially throughout the experiment. Don't worry about how it accelerated.

 

If you want to have the train switch inertial frames, I think you are going to have to factor in the details of relativity of simultaneity, and you may need to decide on a few more details than you're giving.

 

I think it's fairly common that people want to figure out one aspect of relativity that they don't get, and they completely avoid another aspect like it's too complicated to consider. It's like trying to figure out how 2+3=5 without considering the 3... "how does 2 add up to 5, relativity makes no sense!"

Edited February 14, 2015 by md65536

[gib65]

Just to be clear why do you assert that the gap changes?

It's the answer I got to the question in the OP:

For an observer in another frame, the length contracts, which would include the length of space between them. if the trains were 100m long and 100m between them, and they moved such that gamma=2, then each train would be 50m and there would be 50m between them.

So just to be clear, I'm thinking of it like this:

||||||......||||||

The |||||| are the trains and the ...... is the track. If this is what the trains look like when they are at rest, then my understanding is that this is what happens when they start accelerating at the same time at the same rate:

|||||.....|||||

||||....||||

|||...|||

This is what it would look like from the point of view of someone not moving relative to the track. Is this wrong?

(Note that the last ... is not the track contracting, just the space between the trains.)

If you force various points to remain synchronized in the track frame What do you mean by points being synchronized?, the distances between those points will remain the same in the track frame. Effectively you'll be stretching the distance between points (the gap will be increased in the moving train's rest frame) to exactly counteract length contraction. This is set up like Bell's paradox. The answers to your questions can probably be found in an explanation of the paradox. I read some of the wiki article on Bell's Paradox. It says that due to the relativity of simultaneity, the distance between the trains (or rockets in the original formulation) will increase from the train's reference frame because from the train's reference frame, the front train begins accelerating first followed by the rear train. Is this what you're talking about?.

If you don't want to deal with the details of relativity of simultaneity, just give the train a fixed rest length, and let it remain moving inertially throughout the experiment. Don't worry about how it accelerated.

So if the trains are moving at a constant and equal velocities, it wouldn't matter how much they or the gap between them are contracted, the rear train would not look like it was catching up to the front train. Is this right?

What does it mean for something to "move inertially"?

If you want to have the train switch inertial frames, You mean accelerate? I think you are going to have to factor in the details of relativity of simultaneity, and you may need to decide on a few more details than you're giving.

Yeah, I figured the relativity of simultaneity figures into this somehow--if train Tn1 is length contracted, what that means is that the rear of Tn1 is reaching points on the track sooner in the track's reference frame than it is in the train's reference frame (or the front is reaching it later...).

But still, I can't shake the odd conclusion that when both trains are on the track together and accelerated at the same time at the same rate (from the track's reference frame), it will appear that A1 > A2, whereas there would be no reason for this to be true (from the track's reference frame) if the trains were accelerated on different occasions.

I think it's fairly common that people want to figure out one aspect of relativity that they don't get, and they completely avoid another aspect like it's too complicated to consider. It's like trying to figure out how 2+3=5 without considering the 3... "how does 2 add up to 5, relativity makes no sense!"

 

I'm probably making this mistake in some way. My problem is I don't understand relativity well enough and my brain isn't used to thinking about it in the proper way.

 

[Janus]

Here's something that confuses me about length contraction:

 

Suppose you had a toy train Tn1 on a straight track at position P1 (at Tn1's center). You turn on the voltage and watch Tn1 accelerate. It accelerates at rate A1 for an amount of time T1, covering a distance D1, and arriving at position P2 (again, at its center). (Note that Tn1 doesn't stop or slow down on or before P2, it just arrives there).

 

Now you remove Tn1 and place another train Tn2 on the track, only this time you position Tn2 at position P3 which is 20 meters ahead of P1 (where D1 > 20 meters). Again, you switch the voltage on. Tn2 accelerates at A2 for time T2, covering a distance D2, after which it arrives at position P4.

 

Now, given that the conditions of the track do not change throughout this experiment and that Tn1 is identical to Tn2, it would be fair to say that A1 = A2 and that if T1 = T2, then D1 = D2 and P2 - P1 = P4 - P3.

 

So far, none of this denies that Tn1 and Tn2 length contracted as they accelerated. We can say that Tn1 contracted by an amount L1 and that Tn2 contracted by an amount L2, and that L1 = L2.

 

But now, you conduct a third experiment. You repeat the previous two experiments in tandem: both Tn1 and Tn2 are placed on the same track at positions P1 and P3 (respectively). The track is turned on. One would expect both trains to accelerate at A1=A2 and if allowed to continue for time T1=T2, then they should each individually cover distance D1=D2 to arrive at position P2 and P4 (respectively). In other words, nothing from the previous two experiments should change just because we allow the two trains to accelerate at the same time.

 

I would expect both to still contract by L1=L2, but the part that has me confused (still) is that the space between them will also contract.

In the way you have the experiment set up, no it won't. Basically you have set up conditions such that as measured in the track frame the distance between the centers of the trains does not change, and the that the centers have equal accelerations as measured in the rack frame. This is different from the situation where the distance between the centers of Tn1 and Tn2 remain fixed in the frame of the trains. It is in this second case where the track frame would see a contraction in the distance between the train's centers.

What this seems to mean to me (from my most likely inadequate understanding) is that, from the point of view of a person standing still watching the trains go by, the Tn1 will look like it is "catching up" to Tn2--either that or Tn2 is "slowing down" (or at least not accelerating as fast) compared to Tn1--they are closing the gap between them, in other words.

 

This is what it would look like, at least, from the point of view of someone standing still watching the trains go by.

 

So, that person would have to measure that A1 > A2.

 

Maybe that's just the case. Maybe that's just what happens according to relativity. But it seems really odd that simply adding an extra train to the track would change the results of the experiment. Something's not right here.

It basically comes down to this. Either the distance between the centers of the train will remain constant in the track frame( as per your original set up) and will not remain constant in the train frame( neither will either train conclude that the acceleration of the other train is equal to his) or the distance remains constant in the according to the trains but not so according to the tracks.

 

The acceleration of the trains will also be different as measured from each frame.

 

I've cautioned you on this before. Dealing with acceleration in SR can be very tricky and there are a lot of things that can trip you up unless you already have a strong grasp of SR when dealing with strictly inertial frames.

[md65536]

Edit: I agree with Janus' reply, some of my post is redundant. Figuring it all out first using only inertial frames, without specifying the timing of the train's acceleration, would probably be helpful.

 

It's the answer I got to the question in the OP

That answer applies to the case where the train is accelerated in such a way that the rest length of the train remains the same after acceleration.

You have the trains remaining the same only in the track frame.

Someone else had the backs of the trains synchronized.

All of these situations are possible. People will fill in their own details and interpretations unless everything is unambiguously specified.

 

 

So if the trains are moving at a constant and equal velocities, it wouldn't matter how much they or the gap between them are contracted, the rear train would not look like it was catching up to the front train. Is this right?

What does it mean for something to "move inertially"?

If they're moving in unison in a particular frame, they will maintain their distance in that frame. Remember you have to specifically coordinate them to do this, and it is not always physically possible (Bell paradox; a rope or train that can't be infinitely stretched will break at a high enough gamma).

 

Note: It is only the lengths between synchronized points that remain the same. If you synchronize all points (fronts and backs of both trains all synchronized) in the track frame then the trains and gap remain the same length (but must be physically stretched). If you synchronize only the middle points (as I think you might have specified) and let the trains maintain their rest length (this was not specified, so is left open to different configurations), then the trains will contract and the gap will indeed become greater in the track frame.

 

Move inertially, I mean maintain constant velocity, or uses only one rest frame. Yes switching frames means changing relative velocity or accelerating.

 

 

Yeah, I figured the relativity of simultaneity figures into this somehow--if train Tn1 is length contracted, what that means is that the rear of Tn1 is reaching points on the track sooner in the track's reference frame than it is in the train's reference frame (or the front is reaching it later...).

Err... I don't think this line of reasoning will help you until you've figured some other things out first.

 

Frames don't have a single clock. Clocks measure proper time (the time at the clock, not the time at other locations). An event is a single point in space and time, and has a definite proper time according to a clock that passes through the event. A point of the train passing through a certain point on the track is an event, and all observers agree on the proper time that that happens. Now, you can have other clocks on the train and on the tracks, and different observers can disagree on which clocks are ahead or behind relative to the proper time of the given event. But again to say which is ahead or behind I think you'll have to specify the details of how you've coordinated the clocks... and where those clocks are located relative to the event.

I'll work through an example to help myself figure out what I'm talking about...

 

Suppose a train is 100m long in its rest frame and is heading East, and passes a station that's 100m long in the track's frame.

Have 3 clocks on the train, front middle and back, synchronized in the train's frame.

Have 3 clocks at the station, West end, middle, and East end, sync'd in the track's frame.

 

Say the middle clocks pass each other at exactly 12 noon according to the station's middle clock, and 10:00 according to the train's middle clock (edited to avoid introducing misconceptions).

 

According to the train, all 3 clocks on the train read "10:00". The station is contracted, so the front of the train has already passed the East end of the station, and the back of the train has not yet reached the West end of the station.

 

According to the station, all 3 clocks in the station read "12:00" at the moment the middle clocks pass. The train is contracted, so the front of the train has not yet passed the East end of the station, and the back of the train has already passed the West end of the station.

 

To sort out this situation, note that the front clock of the train reads 10:00 only after it has passed the East of the station. According to the station, this hasn't happened yet: the train's front clock does not yet read 10 and is running behind. Similarly, the train's rear clock had struck 10 earlier before reaching the station; it is ahead.

 

Symmetrically, according to an observer on the train, the station's West clock is behind, and the East clock is ahead. Eg. the West clock reads 12 some time after the back of the train has passed, and that hasn't happened yet.

Edited February 14, 2015 by md65536

[gib65]

Thanks Janus and md for answering my questions (the world is right again for me ).

 

 

 

People will fill in their own details and interpretations unless everything is unambiguously specified

 

 

 

I've cautioned you on this before.

 

I hope my questions aren't offensive to you guys. I'm not a scientist and my understanding of this stuff is at an amateur level. Your answers do help though, so I'm thankful for that.

[md65536]

I hope my questions aren't offensive to you guys. I'm not a scientist and my understanding of this stuff is at an amateur level. Your answers do help though, so I'm thankful for that.

I think most people who post here are amateurs, as I am. I don't think anyone is offended by questions about trying to understand relativity (as long as they don't involve purposefully ignoring answers people have already given and don't include "Relativity must be wrong, because..."). Also, no one understands it perfectly or has considered every aspect, so questions can be helpful to everyone. Edited February 18, 2015 by md65536

[swansont]

I don't think anyone is offended by questions about trying to understand relativity (as long as they don't involve purposefully ignoring answers people have already given and don't include "Relativity must be wrong, because..."). Also, no one understands it perfectly or has considered every aspect, so questions can be helpful to everyone.

 

Agree. This is pretty much the whole point of having a discussion board.