Undergrad Proper (and coordinate) times re the Twin paradox

[PeterDonis]

One end lines up the 3.0 meter mark on my x-axis, the other end lines up with the 4.2 meter mark on my x-axis.

To make these observations, what you are calling your "x-axis" has to be a physical object, not a mathematical construct. In other words, you are using "x-axis" to mean, not the mathematical construct that is usually meant by the term "reference frame", but something like what Dale means by "meter stick".

 

[Grimble]

Hmmm.
A reference frame is like a map of Spacetime seen from a particular(the one at rest at the null point of that reference frame) observer's perspective.
I measure 1 hr on the clock I am holding between event 1 and event 2.
You are traveling at 0.6c relative to me and can read that measurement on my clock.
You measure the interval between event 1 and event 2 to be 1.25 hours.
Both measurements are

... invariant.

and both are correct.
The difference is that the clock is measuring time according to my reference frame and you are measuring a larger time according to your reference frame.

Is that measuring 'with' the reference frames; 'relative to' the reference frames; 'against' those reference frames...
If we carry on like this we can forget about science and just debate symantics...

 

[Nugatory]

You measure the interval between event 1 and event 2 to be 1.25 hours.

No, I don't measure that interval. I can't, because those events are not on my worldline.

My measurement is of the interval between two other events (call them event 3 and event 4) which happen "at the same time" as events 1 and 2, and which are on my worldline.

 

[Dale]

I measure 1 hr on the clock I am holding between event 1 and event 2.

That is frame invariant. It is true in any reference frame.

You measure the interval between event 1 and event 2 to be 1.25 hours.

You measure the interval to be 1 hour also. The coordinate time or duration is 1.25 hours in your frame.

The outcome of any measurement can be analyzed in any reference frame. But the quantity measured may not have the same significance in other frames. In your example, Everyone will agree that the time you measured was 1.25 hours, but due to time dilation and the relativity of simultaneity, they will not generally agree that the measurement correctly measured the duration between the two events.

Quantities that are described as "invariant" go beyond this. Not only do all reference frames agree that you obtained the value you did, they also agree that it is the correct value. The spacetime interval is one such measurement, all frames agree on it and agree that it correctly represents the interval. In this case the interval is 1 hr for all reference frames.

Is that measuring 'with' the reference frames; 'relative to' the reference frames; 'against' those reference frames..

You make measurements with physical devices. Reference frames are mathematical conveniences. You don't make measurement with, against, or relative to them. They are part of the analysis, not part of the measurement.

f we carry on like this we can forget about science and just debate symantics..

While that is true, it is important to understand the semantics since they include words with precise technical meanings intended to convey important concepts.

Here, I have repeatedly explained this concept of reference frames being part of the analysis and you are still talking about them being part of the measurement and thinking that the debate is over the use of "with" or "against". Beyond the mere semantics, you are missing the actual underlying point I am making.

 

[Ebeb]

You make measurements with physical devices. Reference frames are mathematical conveniences. You don't make measurement with, against, or relative to them.

I'm not sure I understand the difference between reference frame and measuring device, because if one uses f.ex. the reference frame of a moving train, we (might as well) consider infinite set of synchonized physical clocks being at rest in that frame, and a physical measuring stick of infinite length, at rest per that train. Hence nitpicking about whether reference frame or physical object is measuring seems trivial?

 

[Dale]

we (might as well) consider infinite set of synchonized physical clocks

The clocks are physical, but the synchronization is not. It is a convention.

 

[Ebeb]

The clocks are physical, but the synchronization is not. It is a convention.

Yes, but that doesn't make the row of clocks only an abstract mathematical measuring frame instead of a physical mesuring device.
The synchronized set of clocks, whatever synchronization you use, is stil a physical measuring device.
Same issue with a meter stick. Whatever synchronisation you use for the 'stick events', the set of simultaneous stick events refer to specific physical states of the stick points, hence a physical stick, not an abstract measuring frame.

 

[Nugatory]

The synchronized set of clocks, whatever synchronization you use, is stil a physical measuring device.

Each individual clock is measuring proper time along its worldline, but that is the only physical measurement coming out of the ensemble of clocks. Any comparison between the clocks, any interpretation of what's going on based on the the relationship between the clocks, is based on the convention used to synchronize the clocks.

 

[Ebeb]

Each individual clock is measuring proper time along its worldline, but that is the only physical measurement coming out of the ensemble of clocks. Any comparison between the clocks, any interpretation of what's going on based on the the relationship between the clocks, is based on the convention used to synchronize the clocks.

Yes, but this means there does not exist any "physical measuring stick" at all...?
This means a row (any set) of simultanous events cannot be considered a "physical object", let alone a "physical measuring stick". This gets very interesting. Would you consider the table in front of you a physical object of simultaneous events, or not?

 

[Nugatory]

Would you consider the table in front of you a physical object of simultaneous events, or not?

The table is pretty clearly a physical object with a world tube of its own, but "physical object of simultaneous events" doesn't make much sense. Different observers moving at different speeds relative to one another will have different notions of which events in that world tube are the top of the table at any given moment.

 

[tophatphysicist]

Revisit the space-time diagrams presented by Ebeb on the previous page. I think the situation is quite clear there.

 

[Dale]

The synchronized set of clocks, whatever synchronization you use, is stil a physical measuring device.

But it is not a reference frame. The reference frame is the mathematical tool used in the analysis, not the physical set of clocks. You could choose to analyze those physical clocks using any mathematical reference frame. And you can change the reference frame arbitrarily after the fact, unlike the physical devices.

Yes, but this means there does not exist any "physical measuring stick" at all...?

No, that is not what it means. What it means is that there are physical measuring devices and there are mathematical abstractions. A measuring stick belongs in the first category and a reference frame belongs in the second. Reference frames are not physical things.

This is not a particularly important point. But if a reference frame were something physical then you would have to build something in the physical world in order to change it. Instead, you can change reference frames on a whim without changing anything other than the mathematical analysis. Therefore, a reference frame is part of the mathematical analysis, not part of the physical world.

Think about it. How do you change reference frames? With the Lorentz transformation. What physically changes when you do a Lorentz transformation? Nothing, that is the whole point of the first postulate.

 

[Ebeb]

The table is pretty clearly a physical object with a world tube of its own, but "physical object of simultaneous events" doesn't make much sense.

This sounds contradictory. Because when you state that <<The table is pretty clearly a physical object>> you do consider a set of simultaneous events of the the table.

Different observers moving at different speeds relative to one another will have different notions of which events in that world tube are the top of the table at any given moment.

Indeed. The table they consider will be a different 3D section through the bunch of table atoms worldlines. I don't see why all of a sudden such a different set of simultaneous events would be less physical than your initial 3D table (which you do consider being a physical object). There is no preferred frame.

 

[Ebeb]

This is not a particularly important point. But if a reference frame were something physical then you would have to build something in the physical world in order to change it. Instead, you can change reference frames on a whim without changing anything other than the mathematical analysis.

When you change the reference frame you consider different events being simultaneous. That means the "bomb explodes" and "the dog sleeps" are both part of one 3D physical world as it exist 'now' per one frame. But it will be different per another frame. Another frame means another 3D world of simultaneous events, hence a different 3D physical world where "the dog sleeps" and "the bomb" explodes are not part of one 3D 'now' world.

If you don't consider a different frame = determining a different 3D world, then you cannot even consider one single 3D world being a "physical object", because any 3D world is defined by frame of simultaneity.

 

[Dale]

then you cannot even consider one single 3D world being a "physical object",

I agree, I would not do that, for essentially the reasons you outlined above.

 

[Dale]

Because when you state that <<The table is pretty clearly a physical object>> you do consider a set of simultaneous events of the the table.

I don't know about you, but I consider my table to have not only a height, width, and length (spatial extent), but also a duration (temporal extent). I don't walk into the dining room every morning surprised to find a new table with the same appearance as the old one. I am pretty confident that the table is the same physical object as the one I originally bought*, at least nobody has shown me any physical evidence to the contrary.

*I don't think I would have paid as much if it was only going to last for an instant, regardless of whose simultaneity would be used to define that instant

 

[Bartolomeo]

I am pretty confident that the table is the same physical object as the one I originally bought*, at least nobody has shown me any physical evidence to the contrary.

"Any simple idea is approximate; as an illustration, consider an object, … what is an object? Philosophers are always saying, “Well, just take a chair for example.” The moment they say that, you know that they do not know what they are talking about any more. What is a chair? Well, a chair is a certain thing over there … certain?, how certain? The atoms are evaporating from it from time to time—not many atoms, but a few—dirt falls on it and gets dissolved in the paint; so to define a chair precisely, to say exactly which atoms are chair, and which atoms are air, or which atoms are dirt, or which atoms are paint that belongs to the chair is impossible."
http://www.feynmanlectures.caltech.edu/I_12.html

 

[tophatphysicist]

I think Ebeb is on to something here with his understanding of 3-D frames as cross-sections of a 4-D physical space. Bartolomeo points out the continual changing of the individual particles making up an object. We can understand a collection of 4-D fibers (making up an object) strung out along the 4th dimension with some fibers peeling away while other new fibers join, while still maintaining a 4-D object. And of course observers whose 4-D fiber bundles are oriented differently from each other (different speeds) will observe different 3-D cross-section views of the 4-D object.

I'm sure when Dale commented about expecting his table to still be there the next morning he had in the back of his mind that he expected the 4-D table to extend for many billions of miles into the 4th dimension--he anticipated another continuous sequence of 3-D cross-sectional views of the table as he raced along the 4th dimension at the speed of light.

 

[tophatphysicist]

I don't want to be too presumptuous, but I think Ebeb counts on nature to keep him living in a continuous sequence of reference frames which are consistent with Special Relativity as he moves along his world line. In that context I suspect that he regards each 3-D cross-section of the 4-D space as his reference frame -- as a physical 3-D space -- as an instantaneous cross-section space of the 4-D space.

And I certainly would not want to take anything away from Dale's ability to represent this space mathematically.

 

[Dale]

@Ebeb @tophatphysicist @Bartolomeo

I am not sure if any of the most recent posts are supposed to be relevant to the question of whether a reference frame is mathematical or not. I am also not sure why this is such a hard idea. Measuring devices are physical, reference frames are mathematical. I don't get why this is even a point of discussion.

 

[tophatphysicist]

Of course a 3-D cross-section of 4-D space can be represented mathematically. I agree -- it really shouldn't be necessary to discuss that. Some might use the term, "reference frame" to indicate a particular coordinate system while others might use the term to represent the physical space they are talking about. Perhaps some would suggest that physics is not interested in the existence of a 4-D physical space populated by 4-D world lines (and instantaneous 3-D cross-sections of that 4-D space) . Perhaps some would feel that the concept of physical space is too philosophical and not relevant to physics. In that case this might be a good place to discontinue the discussion.

 

[Ebeb]

I agree, I would not do that, for essentially the reasons you outlined above.

You talk physics, full of physical devices, but you won't accept a 3D world as it exists now (=collection of simultaneous events) is a 'physical object ? Do I understand you correctly?
This would actually mean you don't accept there is a physical 3D world existing around you?

I would agree a frame is a mathematical instrument used to find out what exists 'now', per that frame. I think the reason this is so difficult to grasp is that, -and now I get back to Grimble's initial problem in this thread topic- different frames also show (f.ex) different "coordinate time" between two specific events (two different physical clock displays). That's where people think frames are "only a mathematical tool", because obviously the display on the two clocks won't change. But the clock displaying coordinate time is actually also a proper time display. A frame deals with physical clock displays, proper times displayed on the clock. Hence such a 3D frame shows a physical 3D World. That's what is shown on my diagrams (post#124 and #127).

So, yes, a frame is mathematical thing, and yes you can switch frames as you like, but no, that would not mean that you "change" a physical 3D world. What one does is select a different physical 3D world that exists 'now'. A 3D world is relative, 4D spacetime is invariant. That's what Einstein meant by his 4D existence quotes:

From a "happening" in three-dimensional space, physics becomes, as it were, an "existence" in the four-dimensional "world".

Also:

Since there exists in this four dimensional structure [space-time] no longer any sections which represent "now" objectively, the concepts of happening and becoming are indeed not completely suspended, but yet complicated. It appears therefore more natural to think of physical reality as a four dimensional existence, instead of, as hitherto, the evolution of a three dimensional existence.

 

[Ebeb]

@tophatphysicist,
Yes that's what I have in mind. Glad to the see you are on the same wavelength.

 

[tophatphysicist]

Good commentary, Ebeb. And I was particularly impressed with your space-time diagrams (124 & 127). It presents so clearly what you are saying. I had forgotten that particular quote of Einstein's. It's good to be reminded from time to time.

 

[PeterDonis]

I think Ebeb counts on nature to keep him living in a continuous sequence of reference frames which are consistent with Special Relativity

If that's what he's counting on, he should be very disappointed, because nature doesn't do that. First, the spacetime he's living in is curved, so there are no global reference frames which are consistent with SR. So the "3D worlds" he is talking about--global surfaces of simultaneity in the SR sense--do not exist.

Second, in curved spacetime, there are no 3-surfaces that have all of the properties that "3D worlds" (surfaces of constant time in a global inertial frame) have in flat spacetime. So even if we try to generalize the concept of "3D world" to a curved spacetime, there is no well-defined way to do it. So even in that looser sense, the "3D worlds" he is talking about don't exist.

 

[PeterDonis]

That's what Einstein meant by his 4D existence quotes

You can always put different interpretations on vague ordinary language, even Einstein's. I interpret that quote from Einstein as saying this: relativity tells us that "3D worlds" do not exist; what exists is 4D spacetime. Cutting up that 4D spacetime into slices called "3D worlds" is a human artifact; nothing in the actual 4D spacetime corresponds to it.

If you want to say that your interpretation is more accurate than mine, then you need to find something in the actual 4D spacetime--not just a coordinate choice, but something actual, physical, observable--that picks out a set of "3D worlds" from that 4D spacetime. In flat spacetime, there is a way to do this (by picking global inertial frames); but as I said in my previous post, the special properties that allow you to do that don't exist in curved spacetime.

 

[Ibix]

It's like looking at a cube of metal and arguing about whether it's "really" a set of thin flat plates stacked on top of each other, or stacked on a slope, or corrugated plates or something. It's not "really" anything except a cube. It may be useful to treat it as made up one way or another but, in the absence of annealing marks or whatever, you're imposing a choice.

Dale isn't denying that you can slice the cube. He's just denying that there is One True Way To Slice The Cube.

 

[tophatphysicist]

I knew it would be just a matter of time before someone would bring the discussion into the context of General Relativity. Until we have a theory of unified fields and a theory unifying QM and General Relativity I'm not sure we can make real headway. It is certainly clear that Special Relativity is extremely difficult (if not impossible) to apply in the region of geons and quantum foam. How would you trace out a world line through such chaos? Nevertheless, Special Relativity works quite well in our world. Some would feel that is all we ask of it -- give us a continuous sequence of 3-D cross-sections that extend into the vastness of space that is found to be locally flat, and let it follow the Special Relativity description locally. The far distant black holes do not need to take from us the world we live in.

We could delve more into the details of the bizarre details and implications of high space-time curvature, and even there we might be surprised at the survival of observer 3-D cross-sections. Trouble is that we have no experimental data associated with an actual observer in those extreme circumstances.

Another problem is that physics doesn't seem to have a clear definition of the observer and consciousness, the later seeming to be outside the realm of physics at this point.

 

[Dale]

You talk physics, full of physical devices, but you won't accept a 3D world as it exists now (=collection of simultaneous events) is a 'physical object ? Do I understand you correctly?

Yes. Have I not clearly stated that several times already? The 3D world you mention is not experimentally detectable, so not physical.

But the clock displaying coordinate time is actually also a proper time display. A frame deals with physical clock displays, proper times displayed on the clock.

A frame defines coordinates on a 4D open subset of spacetime. A clock defines proper time only on the 1D worldline of the clock. Even for a clock whose proper time matches the coordinate time, the coordinate time is defined at events where the proper time is not. The two cannot be equated.

That's what Einstein meant by his 4D existence quotes

I am not sure why you quoted that. It seems to support my position.

 

[Dale]

Nevertheless, Special Relativity works quite well in our world.

For many purposes, yes. But there are lots of experiments here in our world that it does not cover.

However, my point is not predicated on either SR or GR. My point is that there is no known method of experimentally detecting the 3D world. To me that makes it non physical.