Understanding Relativity of Simultaneity in Resnick, Part II

[zenterix]

TL;DR Summary

I'd like to understand a section of a book I am reading ("Introduction to Special Relativity" by Resnick).

I will create a series of posts that go through snippets of the section "relativistic kinematics" of the book and try to interpret the content in my own words and perform certain calculations that were left for the reader to do.

In a previous post, I investigated a wrong way to synchronize clocks in a single inertial reference frame.

Here is a correct way

Let us imagine an observer with a light source that can be turned on and off (e.g. a flash bulb) at each clock, $A$ and $B$. Let the measured distance betweeen the clocks (and observers) be $L$.

The agreed-upon procedure for synchronization then is that A will turn on this light source when his clock reads $t=0$ and observer B will set his clock to $t=L/c$ the instant he receives the signal. This accounts for the transmission time and synchronizes the clocks in a consistent way.

An alternative would be

A method equivalent to the above is to put a light source at the exact midpoint of the straight line connecting A and B and inform each observer to put his clock at $t=0$ when the turned-on light signal reaches him. The light will take an equal amount of time to reach A and B from the midpoint, so tha this procedure does indeed synchronize the clocks.

A few observations

1) These methods all place clocks at specific locations and synchronize them using signals (namely, light). This is a practical matter: if we could send a signal from one location to another instantly, ie with an infinite speed, then all clocks could be synchronized at the same instant. We don't have such an infinite-speed signal, however.

2) Given that all clocks in a reference frame can be synchronized in, we can judge time order of events in that frame. The time of an event is measured by the clock whose location coincides with that of the event.

3) Events occurring at two different places in the frame are called simultaneous when the clocks at their respective locations record the same time for them.

Now, the next big question is: suppose one inertial observer does find that two separated events are simultaneous; will these same events be measured as simultaneous by an observer on another inertial frame moving with speed $v$ with respect to the first?

An example to understand relativity of simultaneity
We have two reference frames, S and S', having a non-zero relative velocity.

Two events occur in each frame and leave a mark at that position in space (the book uses as an example a lightning bolt or an explosion of dynamite as an event).

Assume that afterwards, by measurements, each inertial observer finds that he was located exactly at the midpoint of the marks which were left on his reference frame.

Pictorially,

The origin of both frames when the events happen is this midway point. Frame S' is moving to the right relative to frame S with velocity $\vec{v}$.

Implicit here seems to be that we have two observers, and each one is at the origin of one of the two frames. Thus, observer S' (located at the origin of frame S') is moving with velocity $v$ relative to observer S. Is this so?

Because each observer knows he was at the midpoint of the mark left by these events, he will conclude that they were simultaneous if the light signals from them arrive simultaneously at his clock. If, on the other hand, one signal arrives before the other, he will conclude that one event preceded the other. Since each observer has a synchronized set of clocks, he can conclude either that the clocks at the marks read the same time when the marks were made (simultaneous case) or that they read different times (non-simultaneous case).

Ok.

Many different possibilities exist in principle as to what the measurements might show.

I suppose this depends on the velocity of each frame relative to the locations of the events?

Let us suppose, for sake of argument, that the S-observer finds that the lightning bolts struck simultaneously.

This assumption contains the assumption then that frame S is at rest relative to the event locations.

The book then shows a few pictures from the point of view of observer S.

Here we see the light waves from event B arriving at observer S' (who has moved a bit to the right)

and here, at a later time, we see the waves from both events arriving at observer S.

at an even later time, the light from event A arrives at observer S'

One thing that I find confusing here is that $A'$ and $B'$ are moving relative to S and at rest relative to S'.

My impression is that the explosions (or lightning bolts) occurred at some specific location in spacetime that is independent of the frame we are using to give that location coordinates. I would have thought that frame B would be moving away from A and A' and moving towards B and B' (before passing these locations and then moving away from them).

After all, the light signals originated at A and B in frame S (which in frame S' are points with velocities $-\vec{v}$.

Okay, moving on.

The signals arrive at 0 at the same time (as measured from the clock at 0) and so A and B are simultaneous in frame S.

The S'-observer, however, finds that event BB' precedes event AA' in time; they are not simultaneous to him.

Why is the book calling the events BB' and AA' now?

I get that the light signals do not arrive at the same time at the S'-observer. In other words, they arrive at the S'-observer at different times (as measured by frame S' clock at 0' when the signals arrive).

But this BB' thing and the moving B' is bothering me.

 

[zenterix]

After rereading the book, here are some facts that I failed to consider.

The observers note that two lightning bolts strike each, hitting and leaving permanent marks in the frames.

Are there four lightning bolts or two in total?

I'm inclined to think there are two lightning bolts in total.

It's not clear what "leaving permanent marks in the frames" means in practice, but in theory it seems to mean that the marks for a frame are at rest relative to that frame.

Imagine that observer S is on a spaceship that is located between A and B, and observer S' is on another spaceship located between A' and B'.

Now, I guess we have to assume that the spaceships can overlap in space.

Then, two lightning bolts strike the spaceships at each end. As the S' spaceship moves right, the marks left on the spaceship remain in the same position relative to that frame.

Is this a valid explanation?

This explains the movement of $A'$ and $B'$ in frame S.

 

[zenterix]

Do whatever you want. It's just that you are not the first to try this approach, which usually leads to frustration and "This forum is so unhelpful!".

In that case, I am a bit less bothered.

Not getting any answer to a posted question is the same result as not posting a question at all.

And actually, I prefer to at least post the question because it forces me to write the thought process out and frequently allows me to find the answer myself.

Perhaps I will test the homework forum to see if I like it. But like I said, these are not "problems", they are literally trying to understand the theory as presented in a book. So far it seems lots of people like SR on this forum and are willing to help.

 

[Dale]

TL;DR Summary: I'd like to understand a section of a book I am reading ("Introduction to Special Relativity" by Resnick).

I will create a series of posts that go through snippets of the section "relativistic kinematics" of the book and try to interpret the content in my own words and perform certain calculations that were left for the reader to do.

Now, the next big question is: suppose one inertial observer does find that two separated events are simultaneous; will these same events be measured as simultaneous by an observer on another inertial frame moving with speed v with respect to the first?

Not in general, no.

TL;DR Summary: I'd like to understand a section of a book I am reading ("Introduction to Special Relativity" by Resnick).

I will create a series of posts that go through snippets of the section "relativistic kinematics" of the book and try to interpret the content in my own words and perform certain calculations that were left for the reader to do.

Why is the book calling the events BB' and AA' now?

I don’t have the book, but that sounds like a notation choice rather than a substantive issue.

 

[hutchphd]

Why is the book calling the events BB' and AA' now

A refers to the event A described in the stationary (unprimed) frame, so what would A' be? B ? B' ? Perhaps an interstitial comma would help? What nomenclature would you prefer? A and B are distinct but their descriptions are not frame invariant.

 

[PeroK]

@zenterix you have some serious misconceptions about reference frames and events that are buried in your long and wordy explanations.

First, sometimes you use location to mean spacetime location and sometimes to mean spatial location. That's part of your confusion.

Second, you allocate a reference frame in which a given event is at rest. This is wrong. An event is a point in spacetime. An object or sequence of events can be at the same spatial location in a given frame - in which case the object is at rest in that frame. But, no single event an be said to be at rest in a given frame.

Personally, I would take a more mathematical approach, where you give spacetime coordinates to events. The mathematics of the relativity of simultaneity is very simple. Even if the full implications are difficult to grasp. But, you shouldn't actually go wrong in the mathematics.

In any case, your first task is to understand the relationship between events and reference frames, mathematically or otherwise.

 

[zenterix]

@zenterix you have some serious misconceptions about reference frames and events that are buried in your long and wordy explanations.

First, sometimes you use location to mean spacetime location and sometimes to mean spatial location. That's part of your confusion.

Second, you allocate a reference frame in which a given event is at rest. This is wrong. An event is a point in spacetime. An object or sequence of events can be at the same spatial location in a given frame - in which case the object is at rest in that frame. But, no single event an be said to be at rest in a given frame.

Personally, I would take a more mathematical approach, where you give spacetime coordinates to events. The mathematics of the relativity of simultaneity is very simple. Even if the full implications are difficult to grasp. But, you shouldn't actually go wrong in the mathematics.

In any case, your first task is to understand the relationship between events and reference frames, mathematically or otherwise.

Yes, I have noticed that these introductory special relativity books aren't rigorous enough for my taste. Not that they have to be terse and only propositions and theorems, but they could define things better. I definitely admit to the loose and imprecise usage of all these terms.

But one look at the books I am reading quickly shows why: the books are too loose and imprecise.

That being said, I went through my OP and searched for all the occurrences of "location". It is true that the word location was used one time to signify "a location in spacetime", and I specifically wrote "location in spacetime".

So, the usage I have in my OP is by no stretch incomprehensible.

 

[Ibix]

@zenterix - I would say the problem is that your approach is making something really quite simple into an abominably complicated multi-thread mess. It's also worth noting that a former mentor here once commented that for learning relativity, literally any textbook is better than Halliday and Resnick. I must say that it seems to be frequently cited here by very confused students, despite a much better reputation for other topics.

The relativity of simultaneity is quite straightforward: I cannot just declare two clocks to be in sync without checking that it's true, and therefore I need a process to check it. For everyday purposes I can get away with just looking at the two clocks, because the error from neglecting light speed delays (etcetera) are tiny compared to my measurement precision. For high precision I need to subtract out the flight time of light from the clocks to my eye, but that means knowing the speed of light, and since all observers will measure light to pass them at $c$ that means that they cannot agree on simultaneity.

The simplest demonstration of that is Einstein's train thought experiment, which shows that two observers in relative motion can observe the same flashes of light and conclude different things about the timing of their emission. Further reasoning along similar lines will get you to the Lorentz transforms.

 

[Sagittarius A-Star]

It's not clear what "leaving permanent marks in the frames" means in practice, but in theory it seems to mean that the marks for a frame are at rest relative to that frame.

A. Einstein described this scenario with a long train passing an embankment.
The left lightning stroke may cause a mark A on the embankment and a mark A' on the train.
The right lightning stroke may cause a mark B on the embankment and a mark B' on the train.

Einstein's book "Relativity: The Special and General Theory", Part I, Section 9 - The Relativity of Simultaneity:
https://en.wikisource.org/wiki/Rela..._I#Section_9_-_The_Relativity_of_Simultaneity

 

[Dale]

Yes, I would like to understand a section of a book I am reading (from a technical perspective).

One problem is that I don’t have the book, and it is likely that others don’t either. So while we can serve as help for technical and conceptual questions, we cannot help with notational or presentation issues. This question seems to be about the notation. Or at least, if there is a conceptual/technical question it is so deeply buried in notational fog that I missed it.

Since we are giving unsolicited advice, here is mine: focus on the concepts rather than the presentation. We will be better able to help that way.

 

[Sagittarius A-Star]

Then, two lightning bolts strike the spaceships at each end. As the S' spaceship moves right, the marks left on the spaceship remain in the same position relative to that frame.

Is this a valid explanation?

This explains the movement of $A'$ and $B'$ in frame S.

Yes. This explains the movement of $A$ and $B$ in frame S'.

 

[PeterDonis]

I asked questions about special relativity.

You are asking questions about one particular presentation of SR in one particular book. That is not the same as SR itself. And if your opinion of the book is this...

I have noticed that these introductory special relativity books aren't rigorous enough for my taste.

...then you might want to consider looking for a source that is more rigorous in its treatment. Many GR textbooks (such as Misner, Thorne & Wheeler, Wald, and Sean Carroll's online lecture notes) cover SR from the standpoint of differential geometry, which means they pay much more attention to matters of rigor (at least at a physicist's level of rigor--if you are looking for a mathematician's level of rigor you would need to look at texts written by mathematicians).

 

[zenterix]

You are asking questions about one particular presentation of SR in one particular book. That is not the same as SR itself. And if your opinion of the book is this......then you might want to consider looking for a source that is more rigorous in its treatment. Many GR textbooks (such as Misner, Thorne & Wheeler, Wald, and Sean Carroll's online lecture notes) cover SR from the standpoint of differential geometry, which means they pay much more attention to matters of rigor (at least at a physicist's level of rigor--if you are looking for a mathematician's level of rigor you would need to look at texts written by mathematicians).

Indeed I should.

Here is the reason I started studying special relativity by the way: I covered the first four chapters of Purcell's Electricity and Magnetism (I went straight to this book and Griffiths because more intro level books were not giving me the rigor I wanted).

Chapter 5 assumed you know special relativity, and chapter 6 about magnetism assumes you know chapter 5.

So I went over to MIT OCW to one of their introduction to special relativity courses and got the books on the syllabus: French and Resnick.

Initially frustrated with the writing I switched to Susskind. Yesterday, I decided to just check out the relativity of simultaneity part of Resnick to see if that chapter was better than the initial chapters.

But yes, watching a general relativity course last night I realized that in a more rigorous course everything is defined as in a math book.

So, just like with electromagnetism, it seems I need to skip the intro books and go to a slightly more advanced and more mathy book, which I will now do.

All this being said, the introductory level books are sometimes very good at building general intuitions and insights.

A good example of this is linear algebra. Compare a book like Strang's introductory Linear Algebra book with a focus on matrix operations to a book like Axler's Linear Algebra Done Right. Axler is much more rigorous and is a great book but Strang's book is also great (even if the looseness can be an issue occasionally, the insights about how to visualize things is extremely valuable).

I was hoping for something similar with these SR books but I guess I'll have to come back to them later after the more rigorous books.

 

[zenterix]

@zenterix - I would say the problem is that your approach is making something really quite simple into an abominably complicated multi-thread mess. It's also worth noting that a former mentor here once commented that for learning relativity, literally any textbook is better than Halliday and Resnick. I must say that it seems to be frequently cited here by very confused students, despite a much better reputation for other topics.

The relativity of simultaneity is quite straightforward: I cannot just declare two clocks to be in sync without checking that it's true, and therefore I need a process to check it. For everyday purposes I can get away with just looking at the two clocks, because the error from neglecting light speed delays (etcetera) are tiny compared to my measurement precision. For high precision I need to subtract out the flight time of light from the clocks to my eye, but that means knowing the speed of light, and since all observers will measure light to pass them at $c$ that means that they cannot agree on simultaneity.

The simplest demonstration of that is Einstein's train thought experiment, which shows that two observers in relative motion can observe the same flashes of light and conclude different things about the timing of their emission. Further reasoning along similar lines will get you to the Lorentz transforms.

I agree with this.

Except that the point is my question, as I state at the beginning, is to understand precisely this particular book.

In fact, the title of the question is "Understanding relativity of simultaneity in Resnick".

So, the goal is not only to understand the concept, but specifically in the mess that is presented in Resnick.

But it's good to know that the book is considered awful by others.

 

[PeterDonis]

the point is my question, as I state at the beginning, is to understand precisely this particular book

We understand that is what you are trying to do. We are just wondering what the point is, since you already agree that the book is not rigorous enough for you.

 

[zenterix]

We understand that is what you are trying to do. We are just wondering what the point is, since you already agree that the book is not rigorous enough for you.

The conclusion that it is not rigorous enough comes about after reading part of the book, trying to think it through, and asking a question about it.

Still, the authors were trying to expound something in the snippets I quoted.

What the point is is kinda of irrelevant.

Someone could also ask me what the point of studying special relativity is. Or what is the point of studying at all.

Not their concern.

I can use my brain as I see fit. I have a fundamental problem with the prescriptive tone of this entire dialogue. I can study however I'd like. You can choose to answer or not whatever questions you want to.

 

[PeterDonis]

What the point is is kinda of irrelevant.

It might be for you, but that doesn't mean it is for PF. PF is not just about one particular thread or one particular user. We have to maintain an overall signal to noise ratio. At some point, discussion of something that even the OP admits doesn't meet their desired standards becomes noise.

 

[PeterDonis]

the authors were trying to expound something in the snippets I quoted.

That's the usual assumption for a textbook, yes. But sometimes the only way to really figure out what that something is is to find another book that does a better job of expounding it.

 

[zenterix]

That's the usual assumption for a textbook, yes. But sometimes the only way to really figure out what that something is is to find another book that does a better job of expounding it.

This may indeed be true.

And if the answer is "the book is too confusing to be worth the time" then I accept that. That is different than "why are you asking these questions about this book", because the questions come way before reaching the point where one realizes that the book isn't worth the time.

 

[PeterDonis]

Thread closed for moderation.

 

[berkeman]

After a Mentor discussion, this thread will remain closed. There is no conceptual content at this point. The only remaining question is about the notation in one specific book. Thank you to all who tried to help the OP.