The Debate Around Arthur Beiser's Modern Physics Theory

[I_am_learning]

Arthur Beiser in His Modern Phyics books, says that there is absoluteness of past and future. If Event 1 occurs after event 2 in a given frame of reference, Then same sequence occurs in very other frame of reference.
Thats not always true, isn't it?

 

[Mentz114]

It is true if event 1 and event 2 are in the same frame of reference.

Suppose I ring a bell and then walk across my laboratory and ring another bell. Every frame of reference will agree on the order of the bells ringing. The two events mark a section of my worldline. From other frames my worldline is just rotated - not turned upside down.

 

[I_am_learning]

I think that's only true if event 1 and event 2 are separated by large enough time and distance that allows them to be causally related.

 

[Rasalhague]

This answer refers only to inertial frames in special relativity. I don't know enough yet to comment on other cases.

It won't necessarily be true of an arbitrary pair of events that they'll have the same order in all frames. Specifically, if the separation between them is spacelike (i.e. it isn't possible for a signal to be present at both), and we know that E1 precedes E2 in one frame, it will always be possible to chose a frame in which that order is reversed.

But if the separation between a pair of events is timelike (i.e. it's possible for a particle with mass to be present at both, it's possible for them to lie on the worldline of a massive particle), and E1 comes before E2 in one frame, then that order will be the same in all frames, although the time between them may differ. Similarly, if the separation is lightlike (i.e. they both lie on the potential worldline of a pulse of light), their order will be the same in all frames.

I've seen the terms "absolute past" and "absolute future" used as names the past and future nappes of a light cone and the regions inside them.

 

[Rasalhague]

I think that's only true if event 1 and event 2 are separated by large enough time and distance that allows them to be causally related.

That's right.

It is true if event 1 and event 2 are in the same frame of reference.

I've been thinking of events as points in spacetime. From that point of view, a given event exists in all frames, although it may have different coordinates in each frame.

 

[Fredrik]

Rasalhauge: Good answers. (That's an "approve" smiley btw. That's not exactly obvious from the expression on its face). Regarding the terminology "absolute past" and so on, I like the terms "chronological future" and "causal future". I found those terms in a book that someone linked to in another one of these threads. Unfortunately the book used the term "future horismos" for the boundary of the causal future. Maybe it's a good term, but it's a word I've never heard before. What makes it even worse is that its plural form appears to be "horismoi".

Some math nerds don't like the term light cone, because "cone" is a mathematical term, and while both the chronological and causal futures are cones in that sense, their boundary is not. I'm undecided myself. I don't want to start calling it "the future and past horismoi" and be the only one who understands what I'm saying.

thecritic: I think you should have said "separated by a small enough time and large enough distance to make sure that they're not causally related".

Mentz114: You should reconsider the terminology you're using. It really doesn't make sense to say that an event or an object "is" in a frame. (I think we have discussed this before, but it's possible that I remember that wrong).

 

[Mentz114]

Frederik,

It really doesn't make sense to say that an event or an object "is" in a frame.

What does the question below mean ?

Arthur Beiser in his Modern Phyics books, says that there is absoluteness of past and future. If Event 1 occurs after event 2 in a given frame of reference

I assumed that Arthur Beiser means the same as I do by 'frame'.

The scenario I described takes place in my laboratory which is an IFR. Times are measured on a clock comoving with the lab and distances and positions are measured using my own set of rectilinear coords that are carried with the lab along it's worldline. The clock and the coords are the frame.

 

[Fredrik]

His statement is OK. He's just saying that the time coordinate assigned to event 1 by the given frame (=coordinate system) is greater than the time coordinate that the same frame assigns to event 2.

You said "if event 1 and event 2 are in the same frame". Maybe you meant "if the assignments of time coordinates to events 1 and 2 that you're referring to are made using the same frame", or something like that, but what you actually said didn't make much sense. You made it sound like we can have two events, A and B, such that A is in frame S and B is in frame S'. (Each inertial frame in SR assigns coordinates to all events in Minkowski spacetime).

 

[DrGreg]

Mentz, in the quote below

If Event 1 occurs after event 2 in a given frame of reference, then same sequence occurs in every other frame of reference.

the words "in a given frame of reference" are associated with the word "after". The frame is being used to decide the order of events.

It doesn't add any useful information to say "two events are in the same frame of reference", because all events exist in all frames. It makes more sense to say the events are being measured by the same frame of reference.

The quote, as stated by the original questioner, is incorrect without further qualification. It is true if the two events occur at the same place relative to the given frame of reference.

 

[Mentz114]

I can't work out who is supposed to have said what from the above posts.

As far as I'm concerned, 'being in a frame' has a well defined and simple meaning which I've stated ( local clocks and rulers ). AB's statement is correct with that interpretation.

You (plural) must have a different way of understanding of the word 'frame'.

 

[bcrowell]

As far as I'm concerned, 'being in a frame' has a well defined and simple meaning which I've stated ( local clocks and rulers ). AB's statement is correct with that interpretation.

IMO DrGreg has analyzed it correctly in #9.

It's not the idea of "being in a frame" that is bothering us, it's the idea of saying whether two events are or are not in the same frame of reference.

Maybe the issue is that you have a coordinate-dependent representation in mind, whereas we're thinking in terms of a coordinate-independent representation. In a coordinate-independent framework, we think of event E as existing in and of itself, and it could be described in different coordinate systems.

 

[DrGreg]

I can't work out who is supposed to have said what from the above posts.

As far as I'm concerned, 'being in a frame' has a well defined and simple meaning which I've stated ( local clocks and rulers ). AB's statement is correct with that interpretation.

You (plural) must have a different way of understanding of the word 'frame'.

Perhaps you have a different understanding of the term "event" as being a measurement relative to a frame. That's not the usual terminology. If you measure the same occurrence using several different frames, that's still just a single event.

So every event is in every frame, so to specify that an event "is in a frame" tells me nothing I didn't know already.

 

[Mentz114]

IMO DrGreg has analyzed it correctly in #9.

I disagree. I think Fredrik got it right in the first paragraph of #8

It's not the idea of "being in a frame" that is bothering us, it's the idea of saying whether two events are or are not in the same frame of reference.

I think we differ on what 'same' means in this context.

Maybe the issue is that you have a coordinate-dependent representation in mind, whereas we're thinking in terms of a coordinate-independent representation. In a coordinate-independent framework, we think of event E as existing in and of itself, and it could be described in different coordinate systems.

"we think of event E as existing in and of itself". What ?

Let me have one last try. Events can only be described in terms of world lines. If two events are located on a single WL, I say that both events took place in the frame that existed when the WLlines coincided, and their moving frames were the same.

In terms of worldlines the scenario I described is

1. My worldline crosses that of the first bell at which point in spacetime I ring it;
2. ... my worldine separates from the WL of the first bell and continues
3. ... until it intersects the WL of the second clock at which point I ring it.

The clocks are at rest in the lab frame, so we could say the carry the same local frame as the lab.

1 and 3 involve the coincidence of worldlines - and at both events the local frames coincided - so both events are in the lab frame ( which has moved along the lab WL ).
I'm saying that the lab frame does not change as it moves along its WL, and so is always the 'same' frame. And of course the events are coordinate independednt in the sense that they can't be transformed away.


Is this worth arguing about ? I don't follow what you're all trying tell me.

It's getting late but I'll check in tomorrow ...

[edit] 1 minute later ...

So every event is in every frame, so to specify that an event "is in a frame" tells me nothing I didn't know already.

So any two events, which are measured to have precedence A, B from any IFR will have precedence A,B from all IFRs ? Probably not. But if I say that two events are in the same frame as defined above, that tells you something.

 

[Matterwave]

One can have an easy way of thinking of this.

Assume we install a coordinate system (in 1-D, for simplicity) in which there are 2 light bulbs. Light bulb #1 is located at negative 1 light year and light bulb #2 is located at positive 1 light year. You are located at 0. Let's say you observe lightbulb #1 to turn on 1 day before lightbulb 2. Then, if I were an observer located at positive 1 light year (same place as lightbulb #2), I would see light bulb #2 turn on 1 year and 364 days before light bulb #1 turned on. The order is now reversed.

However, if you observe lightbulb #1 to turn on 2 years and 1 day earlier than lightbulb #2 (such that now they can be connected by a light signal), no matter where I put myself, I will always see lightbulb #1 turn on first.

Since the light from lightbulb #1 has reached lightbulb #2 in 2 years (measured by either you or me, since we are both stationary w.r.t. the light bulbs), I can never place myself at some coordinate in which lightbulb #2's light reaches me first. The light-cone of the lightbulb #2 will always be within the lightcone of lightbulb #1.

You can do this more generally for moving observers, but the math gets somewhat more complicated...

You can see from this example that if you accept the principle of relativity (i.e. inertial observers are equivalent and there are no preferred frames), travel at speeds greater than c will cause causality problems.

Say I replaced lightbulb #1 with a gun that shoots projectiles at speed 4c, and this projectile, when it hits lightbulb #2, will turn on lightbulb #2. You will observe the gun go off (looking at the light emitted by the blast), and then you will see lightbulb #2 turn on half a year after that observation. So, to you, the gun causes lightbulb #2 to turn on. However, I will observe the gun go off half a year AFTER I see lightbulb #2 turn on. To me, the effect occurred before the cause. According to special relativity; however, I am just as right as you. This presents a paradox, and is one reason why speeds >c are rejected.

 

[I_am_learning]

thecritic: I think you should have said "separated by a small enough time and large enough distance to make sure that they're not causally related".

Yeah, I was quite wrong. But I think it should have been other way round--> " large enough time and small enough distance to make sure they can be causally related " to make the statement correct i.e. same sequence in every frames.

His statement is OK.

How can that be? We just agreed that a condition needs to be fulfilled for his statement be correct.

 

[Fredrik]

Yeah, I was quite wrong. But I think it should have been other way round--> " large enough time and small enough distance to make sure they can be causally related " to make the statement correct i.e. same sequence in every frames.

That's right. I was a bit sloppy there. I thought we were talking about when the chronological ordering could be different in different frames, but I see now that we weren't.

How can that be? We just agreed that a condition needs to be fulfilled for his statement be correct.

Apparently I was even more sloppy here. Mentz asked me what the statement meant, so I looked at it and saw that Beiser is saying "occurs after event 2 in a given frame". I just wanted to say that these words make sense and that it's clear what they mean. I didn't even notice that it was the statement from #1 that we all agree is wrong.

 

[Fredrik]

Is this worth arguing about ? I don't follow what you're all trying tell me.

I think so. You've been using the words "is in" or "are in" in at least two different meanings, none of which is the same as in "the milk is in the refrigerator" or "1/3 is in the set of rational numbers". So you're using words that have an established meaning to mean something different. It can't be clear to a novice what you mean, and you don't even mean the same thing every time.

This is how you should be thinking: We're using a manifold M to mathematically represent the set of events in the real world. The members of M are also called events. A global coordinate system is a function $x:M\rightarrow\mathbb R^4$. It assigns four numbers $x^\mu(p)$ to each p in M. Those numbers are the coordinates of the event p in the coordinate system x. The word "frame" has a technical definition that I'm not going to go into here. In this context, we can identify the set of frames with a subset of the set of global coordinate systems.

If you'd like to say that an object is at rest in a given frame, then what advantages could it have to completely distort the meaning of the sentence by removing the words "at rest"? And when you say that "if event 1 and event 2 are in the same frame", how can you possibly expect anyone to understand that you mean that "if there's a timelike curve from event 1 to event 2"? It really sounds like you're saying either a) "if there's a frame in which events 1 and 2 are at rest", or b) "if events 1 and 2 are both members of the domain of definition of the same frame".

We can quickly rule out a, because events don't have velocities. So we're forced to go with b, but all frames have the same domain M, so your statement becomes "if events 1 and 2 are members of the set of all events".

I hope that makes it clear enough.

 

[Rasalhague]

Rasalhauge: Good answers. (That's an "approve" smiley btw. That's not exactly obvious from the expression on its face).

Hey, thanks. There's hope for me yet!

Regarding the terminology "absolute past" and so on, I like the terms "chronological future" and "causal future". I found those terms in a book that someone linked to in another one of these threads. Unfortunately the book used the term "future horismos" for the boundary of the causal future. Maybe it's a good term, but it's a word I've never heard before. What makes it even worse is that its plural form appears to be "horismoi".

I guess one good thing about an exotic word is that it probably doesn't have a dozen other meanings in closely related subject areas, or conflicting technical and colloquial meanings, yet... Various meanings in Greek, ancient and modern, but not too hard to google the relativity stuff. These names seem to be quite widespread, transparent and easy to remember: "causal past/future", "timelike past/future", "lightlike past/future", "spacelike elsewhere". I like the mysterious sound of elsewhere.

Some math nerds don't like the term light cone, because "cone" is a mathematical term, and while both the chronological and causal futures are cones in that sense, their boundary is not. I'm undecided myself. I don't want to start calling it "the future and past horismoi" and be the only one who understands what I'm saying.

I was thinking of a cone as a surface, but I see that for Borowski & Borwein (Collins Dictionary of Mathematics) it's only a solid, either finite or infinite, either synonymous with nappe or a "double cone". On the other hand, Wikipedia has solid or surface, and someone here advised me that it could mean either. Wolfram Mathworld calls the cone a kind of pyramid, and a pyramid a kind of polyhedron, and a polyhedron a kind of solid... but adds:

In discussions of conic sections, the word "cone" is taken to mean "double cone," i.e., two cones placed apex to apex. The double cone is a quadratic surface, and each single cone is called a "nappe."

Taylor & Wheeler, in Spacetime Physics, use cone in the sense of a nappe, as defined here. Their past and future light cones of an event are the sets of events connected to that event by a lightlike curve. For chronological or timelike past and future, they use the names "passive past" and "active future" (perhaps chosen, whimsically, to sound like grammatical terms). And the rest is labelled "neutral" or "unreachable" region.

 

[Rasalhague]

The word "frame" has a technical definition that I'm not going to go into here. In this context, we can identify the set of frames with a subset of the set of global coordinate systems.

Are the following synonyms for a homeomorphism from an open subset of M (the underlying set of a manifold) to an open subset of $\mathbb{R}^n$?

(reference) frame
(coordinate) chart
coordinate system

 

[Demystifier]

In my opinion, the concepts of future and past do not make any physical sense without a causal arrow of time which determines a time direction oriented from "causes" to "consequences". On the other hand, the causal arrow of time is merely an illusion emerging from the thermodynamic arrow of time. Thus, in a perfect thermodynamic equilibrium, there is no "future" and "past".

 

[Fredrik]

Are the following synonyms for a homeomorphism from an open subset of M (the underlying set of a manifold) to an open subset of $\mathbb{R}^n$?

(reference) frame
(coordinate) chart
coordinate system

The last two are. I think the first is always the same as the other two if the word "reference" is included. If I remember this right from another thread a month or two ago, MTW defines a "reference frame" as just a coordinate system, and a "proper reference frame" as a coordinate system constructed from the motion and spatial orientation of an object, using the usual synchronization procedure. Mathematically I think the term for the latter is a "Fermi normal chart" (but I'm not 100% sure).

A frame can also be an ordered basis for the tangent space at some point p in the manifold M, or a function $f:\mathbb R^n\rightarrow T_pM$. These two options are pretty much the same because there's a bijection from the set of ordered bases of $T_pM$ into the set of such functions. If $F_p$ is the set of frames at p, then the union $\bigcup_{p\in M}F_p$ is called the frame bundle, and a local section of the frame bundle is called a frame field. So a frame field specifies a basis for the tangent space at each point p in its domain. Some people who say "frame" are referring to a frame field.

I had completely forgotten the term "elsewhere". I like that too.

 

[yuiop]

If Event 1 occurs after event 2 in a given frame of reference, Then same sequence occurs in very other frame of reference.

I do not see why everyone is being so picky about the words used by thecritic. Maybe he should have said something like

"If the coordinate time of event 1 occurs after the coordinate time of event 2 as measured by a given inertial observer or set of inertial observers at rest with respect to each other, the the same sequence of events would happen in the same order as measured by any other inertial observer or set of inertial observers at rest with respect to one another.", but that would probably be too wordy IMHO.

In fact I am finding it hard to find an alternative way of interpreting what thecritic said. Someone please clarify?

It is true if event 1 and event 2 are in the same frame of reference.

I do not see anything terribly wrong with Lut's informal statement although I concede that "It is true if event 1 and event 2 are measured by the same inertial observer or set of observers at rest with respect to one another." might be better technically. My only objection to Lut's statement is that he is clearly implying the order of event's will differ if the event 1 is measured by inertial observer 1 and event 2 is measured by observer 2 who has velocity relative to observer 1. I am not sure why anyone would want to compare events like that and the chronological order of events measured that way would be meaningless.

Thats not always true, isn't it?

Correct. It is not always true. To elaborate on what Rasalhauge and Fredrik have already said, the order of events can appear to be in the opposite order, to observers with relative velocity to each other) if the events are not causally related (i.e. the two events are outside of each other's light cones).

This can be demonstrated by using the Lorentz transformation for time.

$$t' = \frac{(t - vx/c^2)}{\sqrt\left(1-v^2/c^2\right)}$$

If event 1 is at time zero in both reference frames S and S' and event 2 occurs at a later time time t in frame S, then if $x > c^2t/v$, event 2 is measured to occur before event 1 in frame S'.

P.S. I just found this animation http://i121.photobucket.com/albums/o217/Johnny_Hu/gifs/Relativity_of_Simultaneity_Animatio.gif that demonstrates 3 events that happen in the order A>B>C (as measured) in one frame and happen in the reverse order C>B>A (as measured) in another frame. This is because events A and C are outside the future light cone of B. Any event D in the future light cone of event B (light grey triangle above B) would always (be measured to) occur after event B by any inertial observer.