Relativity of Simultaneity and Length Contraction

[curiousburke]

This has probably been discussed/explained many times before so I apologize in advance. Please just direct me to the relevant thread if it has.

In Morin, "Special Relativity For the Enthusiastic Beginner", he explains the loss of simultaneity and specifically the "rear clock ahead" effect. If clocks are synchronized in the "train" frame (L long, v velocity), the rear clock will be ahead by L*v/c^2 when it is simultaneous with the front clock as view from the ground frame.

What I'm wondering, and I have not worked through the math, is whether the time which the back clock is ahead fully explains length contraction as well in that the back of the train would move forward during the time L*v/c^2. Basically, is length contraction just another way of expressing simultaneity effects?

 

[Ibix]

Sort of.

What you think of as a 3d object is a 3d slice along your "now" plane through the 4d object, which is extended in time. With a different simultaneity convention you pick a different 3d slice through the same 4d object. It's rather like deciding to slice a sausage perpendicular to its length or at an angle - the cross section you get is a different shape depending on the angle. In the case of Minkowski geometry, the axis of the 4d object is the line where it's not moving and slices at an angle to this (where it is moving) are shorter in that direction (where they would be longer in Euclidean geometry).

So length contraction is an definitely an effect of your change of simultaneity. But I would not say it's the same thing.

 

[Orodruin]

is whether the time which the back clock is ahead fully explains length contraction as well

It does not. It is however crucial to take relativity of simultaneity into account when deriving length contraction.

If it was just about the relativity of simultaneity, then the lengths would be related by
$$
L’ = L + v\Delta t = L (1 - v^2/c^2)
$$
This is one gamma factor too much!

 

[curiousburke]

It does not. It is however crucial to take relativity of simultaneity into account when deriving length contraction.

If it was just about the relativity of simultaneity, then the lengths would be related by
$$
L’ = L + v\Delta t = L (1 - v^2/c^2)
$$
This is one gamma factor too much!

I admit to trying to do the math and coming up one gamma short (edit: I meant too many). However, thinking about this is hard enough for me that I thought maybe I was missing a gamma due to a frame issue somehow

 

[curiousburke]

Sort of.

What you think of as a 3d object is a 3d slice along your "now" plane through the 4d object, which is extended in time. With a different simultaneity convention you pick a different 3d slice through the same 4d object. It's rather like deciding to slice a sausage perpendicular to its length or at an angle - the cross section you get is a different shape depending on the angle. In the case of Minkowski geometry, the axis of the 4d object is the line where it's not moving and slices at an angle to this (where it is moving) are shorter in that direction (where they would be longer in Euclidean geometry).

So length contraction is an definitely an effect of your change of simultaneity. But I would not say it's the same thing.

So, would you agree with Orodrun that there is an additional contraction above and beyond the clock being ahead?

Sorry, converting between the 4d picture and the 1D train picture doesn't come naturally

 

[Orodruin]

Note that the actual derivation uses two events that are simultaneous in the ground frame and the invariance of the spacetime interval:
$$
\Delta s’^2 = \Delta x’^2 - c^2 \Delta t’^2 = L’^2
= \Delta x^2 - c^2 \Delta t^2 = L^2 (1-v^2/c^2)
$$

 

[Ibix]

So, would you agree with Orodrun that there is an additional contraction above and beyond the clock being ahead?

You need the concept of the interval too, or something of that sort, if you want to derive length contraction starting with the relativity of simultaneity, yes.

 

[curiousburke]

You need the concept of the interval too, or something of that sort, if you want to derive length contraction starting with the relativity of simultaneity, yes.

What do you mean by "the interval"? Is that where the other gamma comes from? :)

edit: Nevermind, spacetime interval. I get what you meant.

 

[Orodruin]

You can also work from the Lorentz transformation directly, but the spacetime interval is more direct and a direct analogy to the Pythagorean theorem in Euclidean space.

 

[curiousburke]

You can also work from the Lorentz transformation directly, but the spacetime interval is more direct and a direct analogy to the Pythagorean theorem in Euclidean space.

Yeah, long pause, I have not gotten into the Lorentz transformation yet. That brings up a question that I have had. Does the Lorentz transformation completely encapsulate SR, specifically does it include everything you need to know about simultaneity?

 

[PeroK]

What do you mean by "the interval"? Is that where the other gamma comes from? :)

edit: Nevermind, spacetime interval. I get what you meant.

Morin covers the relativity of simultaneity (RoS) first, then time dilation, then length contraction. It's good to think for yourself, but if length contraction was the same things as RoS, then why would Morin not show that?

Yeah, long pause, I have not gotten into the Lorentz transformation yet. That brings up a question that I have had. Does the Lorentz transformation completely encapsulate SR, specifically does it include everything you need to know about simultaneity?

Why not focus on the approach in Morin's book? You can't learn everything at once.

 

[curiousburke]

Morin covers the relativity of simultaneity (RoS) first, then time dilation, then length contraction. It's good to think for yourself, but if length contraction was the same things as RoS, then why would Morin not show that?


Why not focus on the approach in Morin's book? You can't learn everything at once.

Well, I thought maybe length contraction was simply a useful way of looking at SR which Morin presents. I know people IRL that claim length contraction isn't "real" and so I was wondering if this was the connection between the two perspectives.

To your second question, like I said, I haven't gotten to Lorentz Transform yet. I'm assuming Morin will get to it. It was just a question that's been rolling around in my head awaiting me to get there or for the info to come from elsewhere.

 

[PeterDonis]

I thought maybe length contraction was simply a useful way of looking at SR

Length contraction by itself is not a useful way of looking at SR. If you are going to use the concept of length contraction, you also need to include time dilation and relativity of simultaneity. All three of them have to be considered together to get a correct picture. Leaving any of them out will mislead you. (The Lorentz transformation includes all three.)

 

[Dale]

Does the Lorentz transformation completely encapsulate SR, specifically does it include everything you need to know about simultaneity?

Yes.

The same can be said of the spacetime interval formula. The Lorentz transform is the transformation that preserves the form of the spacetime interval. And the form of the spacetime interval is unchanged by a Lorentz transform. So you can start with either one and get the other.

 

[curiousburke]

It does not. It is however crucial to take relativity of simultaneity into account when deriving length contraction.

If it was just about the relativity of simultaneity, then the lengths would be related by
$$
L’ = L + v\Delta t = L (1 - v^2/c^2)
$$
This is one gamma factor too much!

I was looking at my chicken scratching, and it still doesn't work out, but wouldn't the time that the clock is ahead (time in the train frame) L*v/c^2 get multiplied by gamma to get the time in the ground frame?

So, it would be:

$$
L’ = L -gamma * L * v^2/c^2
$$

 

[PeterDonis]

I was looking at my chicken scratching, and it still doesn't work out

If you use the full Lorentz transformation formula, it will. That is a key point of my post #13.

 

[curiousburke]

If you use the full Lorentz transformation formula, it will. That is a key point of my post #13.

I wasn't clear. By "doesn't work", i meant that I couldn't get L/gamma from rear-clock-ahead and time dilation.

 

[PeterDonis]

By "doesn't work", i meant that I couldn't get L/gamma from rear-clock-ahead and time dilation.

And if you read my post #13, it will tell you that you should not expect to.

 

[curiousburke]

And if you read my post #13, it will tell you that you should not expect to.

I get that, but I still have to play

 

[PeterDonis]

I get that, but I still have to play

Why?

 

[curiousburke]

Why?

I guess because it goes in the right direction, so I need to prove to myself it it really isn't the same thing.

the back of the trains clock is ahead by Lv/c^2, and the train is short by gamma. It seems a reasonable guess to think they are related

 

[Ibix]

It is all related. The Lorentz transforms are the fundamental bit of maths relating times and lengths in the different frames. Length contraction and time dilation are special cases of the transforms when you make specific assumptions about the quantities you are transforming, and the relativity of simultaneity is a third consequence. That's why the same quantities and combinations of quantities keep appearing in the different formulas.

Sticking special cases together blindly won't get you to the general case, though. Learn the general case, and then you can see how to derive the special cases.

 

[Nugatory]

the back of the trains clock is ahead by Lv/c^2, and the train is short by gamma. It seems a reasonable guess to think they are related

The essential point is that the length of the train is the distance between where its ends are AT THE SAME TIME. Relativity of simultaneity means that train person and platform person have a different notion of AT THE SAME TIME and therefore come up with different lengths.

 

[curiousburke]

It is all related. The Lorentz transforms are the fundamental bit of maths relating times and lengths in the different frames. Length contraction and time dilation are special cases of the transforms when you make specific assumptions about the quantities you are transforming, and the relativity of simultaneity is a third consequence. That's why the same quantities and combinations of quantities keep appearing in the different formulas.

Sticking special cases together blindly won't get you to the general case, though. Learn the general case, and then you can see how to derive the special cases.

I haven't been eager to learn how to apply the Lorentz transforms because I feel like if I understood how to set them up, I would be able to analyze SR problems correctly without understanding the 3 fundamental effects, which is what I'm interested in. I like Morin's development because I think it is leading me to the insight I desire.

 

[Orodruin]

without understanding the 3 fundamental effects

I would certainly not call length contraction, time dilation, and relativity of simultaneity fundamental effects … They are all coordinate artefacts and as such rather uninteresting in themselves. The fundamental idea behind relativity is considering the geometry of spacetime, where these three come in as results of the coordinates we choose to apply.

 

[curiousburke]

I would certainly not call length contraction, time dilation, and relativity of simultaneity fundamental effects … They are all coordinate artefacts and as such rather uninteresting in themselves. The fundamental idea behind relativity is considering the geometry of spacetime, where these three come in as results of the coordinates we choose to apply.

I'm with you on LC and TD, but the relativity of simultaneity seems like a fundamental property of the spacetime that results from c=constant and relativity. Is this just semantics about what is fundamental?

Maybe I'm just enamored with it because it's still new to me.

 

[Orodruin]

I'm with you on LC and TD, but the relativity of simultaneity seems like a fundamental property of the spacetime that results from c=constant and relativity. Is this just semantics about what is fundamental?

If you check my signature you will see that I do consider relativity of simultaneity highly relevant to why people confuse themselves. However, it remains a coordinate effect and as such is not fundamental. I can choose whatever coordinates I wish and choose pick any spacelike surface as a simultaneity.

Perhaps semantics, but I am not about to start calling a coordinate effect fundamental.

 

[Dale]

I like Morin's development because I think it is leading me to the insight I desire.

That is good. I think that any systematic approach is helpful, it is a little less important which systematic approach you use. Of course, it also means that you may have some questions that will be answered later so you will need a little patience.

@Orodruin has a lot of experience teaching this material and he prefers a different systematic approach to Morin's. I happen to also agree with @Orodruin and his approach, but I think that you will do fine with Morin's. Since you are already using it and enjoying it, I think you should continue.

 

[PeterDonis]

I haven't been eager to learn how to apply the Lorentz transforms because I feel like if I understood how to set them up, I would be able to analyze SR problems correctly without understanding the 3 fundamental effects, which is what I'm interested in.

You have this backwards. Lorentz invariance and spacetime geometry are fundamental in SR. Length contraction, time dilation, and relativity of simultaneity are not fundamental effects; they are, as @Orodruin says, coordinate artifacts.

the relativity of simultaneity seems like a fundamental property of the spacetime that results from c=constant and relativity.

Relativity of simultaneity is a coordinate artifact that results from how inertial coordinates are constructed; how they are constructed is indeed constrained by the constancy of the speed of light and the principle of relativity, but that doesn't make coordinate-dependent quantities fundamental.

The real lesson of relativity is that coordinate-dependent quantities have no physical meaning: that includes simultaneity. The things that have physical meaning are invariants: things that don't depend on how you choose your coordinates. Simultaneity is not one of those things.

 

[Orodruin]

@Orodruin has a lot of experience teaching this material

I believe it is now 20 years since I first TAd a relativity course. Coincidentally I also have my 10 year PF birthday coming up a month from now so half that time spent here.

I happen to also agree with @Orodruin and his approach, but I think that you will do fine with Morin's. Since you are already using it and enjoying it, I think you should continue.

The order Morin does things in is very common to be fair. It is also how things are often laid out in popular texts, which tend to emphasize length contraction and time dilation. I prefer the geometrical approach because it goes into the fundamentals first and not the things that tend to just confuse people.

 

[Dale]

I believe it is now 20 years since I first TAd a relativity course. Coincidentally I also have my 10 year PF birthday coming up a month from now so half that time spent here.

Oh, wow. I didn't realize it was that much experience! We are getting old

I prefer the geometrical approach because it goes into the fundamentals first

Me too. I also think that the geometrical approach is more helpful for eventually learning GR where it is unavoidable to my knowledge. But lots of people (including me) learned it starting from time dilation, etc.

 

[curiousburke]

It's been 30 years since undergrad course covering SR for me, and I'm not trying to insist on my definition of "fundamental", but that is the word Morin uses for these three effects.

 

[PeterDonis]

It's been 30 years since undergrad course covering SR for me, and I'm not trying to insist on my definition of "fundamental", but that is the word Morin uses for these three effects.

Yes, understood. Just be aware that that definition of "fundamental" is limited.

 

[Nugatory]

I'm with you on LC and TD, but the relativity of simultaneity seems like a fundamental property of the spacetime that results from c=constant and relativity.

Relativity of simultaneity is every bit as much a coordinate effect as length contraction and time dilation - relativity of simultaneity just says that two different frames can assign different $t$ coordinates to the same two events.

The essential and coordinate-independent property of spacetime that all of these coordinates effects hinge on is the invariance of the spacetime interval.

 

[Dale]

@Orodruin this might be a good spot to link to any material you have published using your approach. It is the second page of an ongoing thread where it is pertinent, so it isn’t really spammy.