Whose Clock Slows Down in Relativity?

[PAllen]

Interesting that you would think that. About a month ago, I exchanged several emails with Steve Gull. I had asked him if he realized that his simultaneity was non-causal. He replied that "The construction is SHAMELESSLY acausal. Probably should have made a point about that...".

Fredrik (who explained the Dolby & Gull simultaneity in one of his posts) also realized that D&G is non-causal, although he didn't consider that a "show-stopper", like I do.

Mike Fontenot

The difference is using it as operational definition in the real world versus an arbitrary coordinate system. For the former, it is strictly causal. You can only define what event in the past of your timeline was simultaneous to an event you have become aware of. The most you can do about events not yet causally related to you is predict based on your imperfect knowledge; for this purpose, it works as well or poorly as your prediction reliability (including predicting your own future). Note that Dolby & Gull did not invent their notion, they just wrote a paper about it that got attention. In my view it is strictly causal because it only operationally maps your past lightcone.

Given a complete spacetime model, you can build a complete coordinate mapping. In such a mathematical construction, time is just a coordinate, all past and future are determinate, so worrying about causality is not meaningful.

 

[Dale]

Fredrik (who explained the Dolby & Gull simultaneity in one of his posts) also realized that D&G is non-causal, although he didn't consider that a "show-stopper", like I do.

It isn't a "show stopper" in any way other than your personal preference. You are certainly not required to use any other coordinate system than your preferred one, but you are flat out, simply, and without reservation WRONG in your claim that your pet system is the only valid one.

 

[Mike_Fontenot]

Previously, I wrote:

Another issue was arbitrary definition (or lack thereof) of what 'elementary measurements and calculations' are.

Those elementary observations and elementary calculations are given, in detail, in my paper. I'm not willing to reproduce them here. But I have provided some hints in some previous posts:

https://www.physicsforums.com/showpost.php?p=2960120&postcount=41

https://www.physicsforums.com/showpost.php?p=2978931&postcount=75 .

To understand why ALL of the other alternative reference frames for an accelerating traveler (other than mine) contradict the traveler's own elementary measurements and elementary calculations, you have to understand two different things:

1) You first need to understand what those measurements and calculations are, for a PERPETUALLY INERTIAL traveler,

and

2) you need to know how to then apply your understanding of item 1 to the case of a traveler who accelerates.

Anyone who has a good understanding of the fundamentals of special relativity should be able to do item 1. In fact, going though the analysis yourself will be much more beneficial to your understanding than just hearing the solution from me or from anyone else. The required analysis IS elementary, but it is easy to get wrong. Keep at it ... you haven't succeeded until your result agrees with the Lorentz equations. The problem to be solved is described here:

https://www.physicsforums.com/showpost.php?p=2960120&postcount=41 .

If you can't get the solution on your own, read my paper.

Item 2 requires that you consider what happens if and when the accelerating traveler decides to stop accelerating, and to remain unaccelerated for some segment of his life, before deciding to accelerate again. It is necessary to ask, and correctly answer, the following two questions:

"When an accelerating traveler stops accelerating, how long must he wait before he can be legitimately considered to be an inertial observer (and thus be able to legitimately use the Lorentz equations to determine simultaneity at a distance)?"

and

"When a here-to-fore inertial observer begins to accelerate, at what point in his life, before he accelerates, does he lose his status as an inertial observer?".

The correct answer is that for ANY unaccelerated segment of the life of the traveler, no matter how short the segment, the traveler is a full-fledged inertial observer during that ENTIRE segment (and he is thus able, during that entire segment, to legitimately use the Lorentz equations to determine simultaneity at a distance).

You may be able to arrive at that answer yourself, if you spend some quality time thinking about those questions. If you can't, read my paper.

Mike Fontenot

 

[bobc2]

Easy this to say. Hard to prove. If you think about it long enough, my bet is that you change your opinion on that matter. The only proof we need in support of said contention is a standard Minkowski diagram, and the assumption that what is predicted by the LTs "is real". It's not in conflict with the LTs in any way. It merely addresses the nature of time, and hence a deeper implication of the nature of spacetime.

Good post, GrayGhost. Let me take a shot at sketching the Minkowski diagram. I'll use the setup with a blue guy moving to the right at relativistic speed with respect to a black rest system. A red guy is moving to the left in that same system with the same speed (blue and red guys moving in opposite directions with same relativistic speed). In the upper left sketch I have marked off equal distance positions along the respective world lines for the blue guy and the red guy. The fact that we use this symmetric spacetime diagram assures that the calibrations of distances for the blue and red coordinates are the same (otherwise it would be necesary to use the hyperbolic calibration curves).

The blue and red guys each reach the position number 9 at proper times that match each other. The rockets and all objects inside the rockets, including clocks and human bodies, are 4-dimensional objects, so it can be perplexing trying to comprehend who or what is doing the moving. The usual language is that each observer moves along his own world line at the speed of light, c. Einstein's colleague, Hermann Weyl, said something like, "...the observer crawls along his own world line." So, some distance traveled along the 4th dimension would be dX4 = c(dt). To avoid a sidebar on that issue, for now, let's just play like there is some aspect of nature associated with consciousness that moves along the 4th dimension at speed c. It is interesting to ponder the enormous length of a life sized 4-D object along its X4 dimension as compared to their almost negligible X1 length.

So far we have a sketch of the R4 manifold (supressing X1 and X2) with the world lines of the 4-D objects. So, we're having really a purely spatial discussion. The X1' coordinate (blue) and the X'' coordinate (red) is oriented such that the photon world line always bisects the angle between X1 and X4, X1' and X4', ...X1'' and X4''. A photon world line would always bisect the angle between any observer's X1 axis and X4 axis. As a result of this circumstance of the 4-dimensional world, every observer will observe a ratio of 1:1 between displacement along the X4 axis and displacement along the X1 axis for any photon world line. In other words, all observers measure the speed of light as c.

Now, having set up the Minkowski spatial picture, we see in the upper right sketch that one can form a right triangle using X4'', X1', and X4'. Given this purely 4-dimensional relationship, it requires only high school math, Pythagorean theorem, to derive the Lorentz transformations. Here, for example, we derive the transformation formula (rotational) for time, t'' (a clock reading in the red rocket), as a function of the blue clock's time, t'. (see lower right sketch)

The formula indicates that the red guy's clock will lag behind the blue guy's clock as observed by the blue guy. The blue guy's instantaneous 3-D cross-section of the universe, blue's "NOW", intersects the red guy's 4-dimensional rocket at red position 8 when the blue guy is at position number 9. This explains quite clearly the reason for the time dilation effect of special relativity.

But, red's "NOW" (when the red guy is at his position number 9) 3-D cross-section intersects blue's 4-dimensional rocket at blue's position number 8, and sees that the blue guy's clock is lagging behind his own.

Likewise, the lower left sketch illustrates the length contraction effect of special relativity.

 

[GrayGhost]

bobc2,

Thanx. Your post looks fine to me as well. As drafted, I do not see any problem with the illustration. However, what was the general point you were intending to convey in your post?

The use of the Loedel figure does speed things up in proving the point regarding the spacetime interval length s, however it works only because of the particular symmetry used in Loedels whereby the fictitious stationary ict bisects the moving worldlines. IOWs, try applying your formulation process to a standard Minkowski illustration with 1 stationary and 1 moving observer, see what happens. If you do, and are successful, you'll find out how Minkowski formulated it and the meaning of the imaginary i.

EDIT: I was mistaken in my highlighted statement. It the formulation works just the same from a standard Minkowski illustration as from a Loedel figure.

GrayGhost

 

[Dale]

ALL of the other alternative reference frames for an accelerating traveler (other than mine) contradict the traveler's own elementary measurements

No they don't. You have never demonstrated this and it is false. All frames will agree on the result of any measurement.

 

[GrayGhost]

The pathlength is the length of the spacetime interval (s), which is nothing more than the proper time experienced by the oberver at both events. Although s is the length of the hypthenuse (slanted worldline of the traveler) on a Minkowski diagram, it is temporally shorter than the stationary observer's vertical time axis (over the same interval).

You have something inverted. Unlike a straight line in spatial dimensions, the interval of a freely falling object in relativity between two spacetime events is not a minimal but a maximal displacement measured in proper time.

)

I missed this one before ...

Incorrect. I was talking SR. He who resides at both events always measures the least amount of time for the interval. If 2 observers reside at both events, eg in the twins scenario, then the one who travels the shorter path thru the continuum ages less over the interval, because he accrues less proper time.

GrayGhost

 

[bobc2]

bobc2,

Thanx. Your post looks fine to me as well. As drafted, I do not see any problem with the illustration. However, what was the general point you were intending to convey in your post?

I was wanting to place emphasis on the aspect of the pure geometric description of the 4-dimensional universe.

The use of the Loedel figure does speed things up in proving the point regarding the spacetime interval length s, however it works only because of the particular symmetry used in Loedels whereby the fictitious stationary ict bisects the moving worldlines. IOWs, try applying your formulation process to a standard Minkowski illustration with 1 stationary and 1 moving observer, see what happens. If you do, and are successful, you'll find out how Minkowski formulated it and the meaning of the imaginary i.
GrayGhost

Yes, I'm familiar with that, and of course you are correct. But, the ict seems not so significant when you simply swap signs in the derived metric, then arbitrarily substitute the (ict')^2 and (ict'')^2 for -(X4')^2 and -(X4'')^2. I can do the same thing with any simple right triangle on a piece of paper without anyone assigning anything mysterious or unusual about one of the lines (legs) of the triangle.

Einstein begins Appendix II of his book, "Relativity - The Special and the General Theory" by stating, "We can charcterize the Lorentz transformation still more simply if we introduce the imaginary ict in place of t, as time-variable... x4 = ict.

 

[GrayGhost]

the ict seems not so significant when you simply swap signs in the derived metric, then arbitrarily substitute the (ict')^2 and (ict'')^2 for -(X4')^2 and -(X4'')^2. I can do the same thing with any simple right triangle on a piece of paper without anyone assigning anything mysterious or unusual about one of the lines (legs) of the triangle.

Yes, however I see I was mistaken ... on the matter of your spacetime interval formulation not applying for normal Minkowski illustrations with 1 stationary and 1 moving observer. It in fact does apply, of course. Thinking one thing, talked about another. It just doesn't work for Loedel's (geometrically per the diagram) if the 2 moving worldlines are not symmetric about the stationary ict axis.

The best formulation of a spacetime interval, is one which presents the y-axis as well ... so an x,y,ict worldline illustration. The 2-d lightcone (versus a ray's lightpath) helps as well. It explains 2 things ... (1) the meaning of imaginary axes, and (2) why the magnitude of s is always less than the magnitude of ct.

GrayGhost

 

[bobc2]

Yes, however I see I was mistaken ... on the matter of your spacetime interval formulation not applying for normal Minkowski illustrations with 1 stationary and 1 moving observer. It in fact does apply, of course. Thinking one thing, talked about another.

You know, I had no idea what a Loedel diagram was. I made an assumption in my mind about the diagram I figured you had in mind (one used for the not-symmetric case when one traveler has motion with changing directions, etc.) and knew that you were correct in commenting that one would not use the symmetric diagram in that case.

It just doesn't work for Loedel's (geometrically per the diagram) if the 2 moving worldlines are not symmetric about the stationary ict axis.

I'm sure you realize that as long as each observer is moving at constant velocity, you can always find a rest system for which you can then use the symmetric diagram (just find the system for which the red guy and blue guy are moving in opposite directions at the same speed). I can do the twin paradox using a symmetric diagram for the trip out. I add in the hyperbola calibration curves, then use those same calibration curves with a second shifted rest system with increased speed for the guy making the return trip. The calibration curves show clearly the shorter 4-D path for the return trip.

I like the symmetric diagram (now I know to call it the Loedel diagram), because you can illustrate all kinds of seemingly paradoxical situations with it, like the pole-in-the-barn example below.

The best formulation of a spacetime interval, is one which presents the y-axis as well ... so an x,y,ict worldline illustration. The 2-d lightcone (versus a ray's lightpath) helps as well. It explains 2 things ... (1) the meaning of imaginary axes, and (2) why the magnitude of s is always less than the magnitude of ct.GrayGhost[/QUOTE]

Yes, I like that presentation as well.

 

[GrayGhost]

Bobc2,

Wrt your illustration at ...

https://www.physicsforums.com/showpost.php?p=3106843&postcount=39"​

When RED is at 9, BLUE is at 8 in RED's sense of NOW (cosmos wide). Yet, that very same BLUE fellow knows that "the only RED that presently exists" is over yonder with a clock readout of 7 (not 9). So whatever RED was doing before (two units of time prior to 9, mayby scratching his chin) is what is presently happening wrt BLUE, and its real, and RED (who is at 9) knows it per the theory. This suggests to RED (and anyone else) that there exists in-the-future a BLUE who holds RED himself presently at 9, and some future RED that holds said BLUE fellow in the present moment. All are correct, as all LT predictions are assumed real and not illusionary effect.

One conclusion that may be drawn is that an infinite number of REDs exist, each occupying some unique moment upon his worldline, all progressing steadily thru his path w/o knowledge of each other. IOWs, you exist everywhere along your lifecycle at once, and each moment of yourself progresses in unison at equal rate unaware of each other. This is what Brian Cox was taking about, an implication of relativity theory when taken at face value. The worldline sits there in the continuum, in its entirety, always.

OK, let the bullets fly :)

GrayGhost

 

[Phrak]

I missed this one before ...

Incorrect. I was talking SR. He who resides at both events always measures the least amount of time for the interval. If 2 observers reside at both events, eg in the twins scenario, then the one who travels the shorter path thru the continuum ages less over the interval, because he accrues less proper time.

I leave your education in the overtaxed hands of the PF staff.

 

[bobc2]

Bobc2,

Wrt your illustration at ...

https://www.physicsforums.com/showpost.php?p=3106843&postcount=39"​

When RED is at 9, BLUE is at 8 in RED's sense of NOW (cosmos wide). Yet, that very same BLUE fellow knows that "the only RED that presently exists" is over yonder with a clock readout of 7 (not 9). So whatever RED was doing before (two units of time prior to 9, mayby scratching his chin) is what is presently happening wrt BLUE, and its real, and RED (who is at 9) knows it per the theory. This suggests to RED (and anyone else) that there exists in-the-future a BLUE who holds RED himself presently at 9, and some future RED that holds said BLUE fellow in the present moment. All are correct, as all LT predictions are assumed real and not illusionary effect.

One conclusion that may be drawn is that an infinite number of REDs exist, each occupying some unique moment upon his worldline, all progressing steadily thru his path w/o knowledge of each other. IOWs, you exist everywhere along your lifecycle at once, and each moment of yourself progresses in unison at equal rate unaware of each other. This is what Brian Cox was taking about, an implication of relativity theory when taken at face value. The worldline sits there in the continuum, in its entirety, always.

OK, let the bullets fly :)

GrayGhost

Good job, GrayGhost. Some philophers/metaphysicists take from the 4-D universe picture that there must be something involving consciousness doing the moving along the world lines. And it has been suggested that there are at least two different models for this: 1) the one you just described (consciousness all along the world line) or 2) one global 3-D consciousness moving in a way that results in a single 3-D becoming experience for the separate observers (this one is particularly grotesque; it results in zombies along the world lines that are not participating in the single 3-D NOW).

Appologies for getting too far off topic and bringing in too many contrived ideas for a physics forum. I must say that I just have no idea what is really going on with time, consciousness, experience, or reality in general, either from the standpoint of ontology or epistemology. I think nature has much mystery yet to be unraveled (if indeed it is ever possible to unravel it). I do think the objective 4-D universe populated by 4-D objects is valuable as a pedagogical tool in understanding special relativity.

 

[Dale]

If you are going to talk about consciousness, ontology, and epistemology, please start a thread in the philosophy section. It does not belong here.

 

[GrayGhost]

bobc2,

Yes, you did bring in the use of philosophy and used the word metaphysics in your prior post. No apology necessary, as I personally have no issue with that. I myself was not addressing philosophy. I was addressing the theory from a science POV, although when discussing implications of the relativity, it always ventures near that gray area between known physics and the metaphysics. One can argue that to discuss the meaning of the theory, over and above the mere usage of existing formulae, is to engage in metaphysics. Yet, wrt my position (and Brian Cox's, as well as many others), I spoke only in relation to the existing accepted transformations. I was not speaking of metphysical topics in which the math has not been developed.

It should also be pointed out that Newton, Faraday, Maxwell, Einstein, Lorentz, and even Hawking did so. Every idea these great minds ever had was metaphysics, "until" they developed the math for their theory AND had it accepted by the leading scientific community at large. I mean if physics forums such as these existed back in 1905, just imagine what would have been thought of a patent clerk who claimed time slowed down relatively, or asked "what would it be like to ride a beam of light?". On the other hand, for every metaphysical idea, the vast vast majority of them are either incomplete or bogus, so I do understand why the forum here prefers such discussions don't happen. I personally do not believe I crossed that boundary in this thread, however I find that you seem as much interested in philosophy as in the physics (which is fine as well). So yes, the discussion went off target it seems, and the forum appears unhappy. The forum here does not wish to engage into the meaning of theories, but instead, only the usuage of the equations. No problem, I can comply.

One last comment on the matter ... when someone asks "are the contractions real or not", no one in this forum can adequately answer that question to satisfaction, because it would have to be deemed "metaphysics" to explain WHY a proper length and moving contracted length can exist concurrently in nature. IOWs, "look at the equations and solns" is the most likely response you'll get. Another typical response ... "because observers are allowed to disagree, given the 2 postulates true". Yet, no one ever takes the time to ponder how nature must be designed if those things are in fact true. It's easier to say ... "don't ask, just use the formulae". Those kind of responses alone pretty much guarantee few will ever grasp the meaning of it, even though they learn to toss the equations around well enough, and gain superficial understanding of the model. This is unsatisfactory IMO, however I'll comply with the forum's wishes. From here out, any posting I do in the physics forum here will possesses nothing but the equations, and the always typical verbal descriptions, to keep everyone happy.

GrayGhost

 

[Dale]

when someone asks "are the contractions real or not", no one in this forum can adequately answer that question to satisfaction, because it would have to be deemed "metaphysics" to explain WHY a proper length and moving contracted length can exist concurrently in nature.

I disagree with your "because". Noone in this forum can adequately answer the question because no one has a scientific definition of the word "real".

That said, your conversation with bobc2 is not really appropriate for this forum. The primary mission of this forum is educational in nature. We are not here to advance science, but rather to teach it. That is why we avoid metaphysics and speculation. Certainly that is valuable for the advancement of physics, but that is the purpose of scientific conferences, publications, graduate schools, and informal gatherings amongst researchers.

 

[GrayGhost]

Incorrect. I was talking SR. He who resides at both events always measures the least amount of time for the interval. If 2 observers reside at both events, eg in the twins scenario, then the one who travels the shorter path thru the continuum ages less over the interval, because he accrues less proper time.

I leave your education in the overtaxed hands of the PF staff.

Phrak,

Wrt all due respect, I'd appreciate it if you could prove your assertion that the magnitude of s is not less than ict, as you contend.

The invariant spacetime interval equation ...

s²=-(ct)²+( x² + y² + z² )​

(ict')²=-(ct)²+( x² + y² + z² )​

-(ct')²=-(ct)²+( x² + y² + z² )​

(ct')²=(ct)²-( x² + y² + z² )​

c = 1

(t')²=(t)²-( x² + y² + z² )​

t²= t'² + ( x² + y² + z² )​

Which models the Pathagorean's theorem. The length one travels thru 4-space is numerically equivalent to proper duration he accrues over the interval. Therefore numerically, |s|=|t'| ... and we know from the above that t' < t, and so numerically ... |t| > |s|

Now if what I say is wrong, I'd like to know what it is, as opposed to the insults. Does it still appear incorrect to you?

GrayGhost

 

[GrayGhost]

DaleSpam,

Fair enough.

Wrt "real" though, I do recognize your point indeed. Yet, if there were no scientific definition of the word real, then physics would not be a science of reality. I'm presuming that you make a distinction of sort between "real" versus "consistently measurable"? Anyways, I'll leave it at that. Thanx for pointing out format and goal of the forum. I'll do my best to comply.

GrayGhost

 

[Dale]

Yet, if there were no scientific definition of the word real, then physics would not be a science of reality.

I have never seen any scientific definition of the word "real", but if you have then I would be glad to learn.

 

[bobc2]

It should also be pointed out that Newton, Faraday, Maxwell, Einstein, Lorentz, and even Hawking did so. Every idea these great minds ever had was metaphysics, "until" they developed the math for their theory AND had it accepted by the leading scientific community at large. I mean if physics forums such as these existed back in 1905, just imagine what would have been thought of a patent clerk who claimed time slowed down relatively, or asked "what would it be like to ride a beam of light?". On the other hand, for every metaphysical idea, the vast vast majority of them are either incomplete or bogus, so I do understand why the forum here prefers such discussions don't happen...

...One last comment on the matter ... when someone asks "are the contractions real or not", no one in this forum can adequately answer that question to satisfaction, because it would have to be deemed "metaphysics" to explain WHY a proper length and moving contracted length can exist concurrently in nature. IOWs, "look at the equations and solns" is the most likely response you'll get. Another typical response ... "because observers are allowed to disagree, given the 2 postulates true". Yet, no one ever takes the time to ponder how nature must be designed if those things are in fact true. It's easier to say ... "don't ask, just use the formulae". Those kind of responses alone pretty much guarantee few will ever grasp the meaning of it, even though they learn to toss the equations around well enough, and gain superficial understanding of the model.

Very nice and reasonable post.

 

[yuiop]

You meant the shortest path thru spacetime, yes? The pathlength is the length of the spacetime interval (s), which is nothing more than the proper time experienced by the oberver at both events. Although s is the length of the hypthenuse (slanted worldline of the traveler) on a Minkowski diagram, it is temporally shorter than the stationary observer's vertical time axis (over the same interval).

Defining path length through spacetime as the spacetime interval is one way of doing things and possibly the formal/regular way, but it was not what I meant. Let us say we plot a right angled triangle with vertices A,B and C on a graph with the axes labelled x and y, with the side AC being the hypotenuse. The path from A to C is longer than the the path from A to B or from B to C. It is by this definition that I mean that the longest path through spacetime (with time substituted for the y dimension) is the hypotenuse and it the longest path that experiences the least proper time. Saying the path with the shortest proper time interval is the path that experiences the least proper time is a circular definition. Defining the longest spacetime path length (with time scaled to an equal footing with space) as the path with the least elapsed proper time works in any number of dimensions. For example if we have a clock on the perimeter of a rotating disk then this clock moves in two space dimensions and one time dimension and traces out a spiral path in the inertial frame that remains at rest with the centre of the disk. This spiral path is longer than the vertical path of a clock that remains at rest in the inertial frame and so the clock on the disk experiences less elapsed proper time. In the classic twins experiment, the twin that travels away from Earth and then returns has the longest spacetime path and experiences the least proper time. When the twins experiment is transformed to the point of view of any inertial observer, it always remains true that the twin with least elapsed proper time travels the longest path through spacetime (in SR).

 

[bobc2]

Defining path length through spacetime as the spacetime interval is one way of doing things and possibly the formal/regular way, but it was not what I meant. Let us say we plot a right angled triangle with vertices A,B and C on a graph with the axes labelled x and y, with the side AC being the hypotenuse. The path from A to C is longer than the the path from A to B or from B to C. It is by this definition that I mean that the longest path through spacetime (with time substituted for the y dimension) is the hypotenuse and it the longest path that experiences the least proper time. Saying the path with the shortest proper time interval is the path that experiences the least proper time is a circular definition. Defining the longest spacetime path length (with time scaled to an equal footing with space) as the path with the least elapsed proper time works in any number of dimensions. For example if we have a clock on the perimeter of a rotating disk then this clock moves in two space dimensions and one time dimension and traces out a spiral path in the inertial frame that remains at rest with the centre of the disk. This spiral path is longer than the vertical path of a clock that remains at rest in the inertial frame and so the clock on the disk experiences less elapsed proper time. In the classic twins experiment, the twin that travels away from Earth and then returns has the longest spacetime path and experiences the least proper time. When the twins experiment is transformed to the point of view of any inertial observer, it always remains true that the twin with least elapsed proper time travels the longest path through spacetime (in SR).

youiop, I think GrayGhost may have been picturing 4-dimensional distances differently from what you describe here. Consider the spacetime diagram below for the situation you were describing. When drawing a spacetime diagram for this situation I think it may help to put in the hyperbolic calibration curves that keep track of the proper distances for the two observers. The "line lengths" that you would measure putting your ruler on the computer screen do not correspond to the actual proper distances; you must use the calibration curves. Proper distance is of course just the speed of light times the proper time.

The sketch of the popular twin paradox is intended to convey exactly what GrayGhost put into words. The thin green lines at 45 degree angles represent photon world lines (they of course travel at speed c). (Sorry if I've misread your meaning)

 

[PAllen]

Defining path length through spacetime as the spacetime interval is one way of doing things and possibly the formal/regular way, but it was not what I meant. Let us say we plot a right angled triangle with vertices A,B and C on a graph with the axes labelled x and y, with the side AC being the hypotenuse. The path from A to C is longer than the the path from A to B or from B to C. It is by this definition that I mean that the longest path through spacetime (with time substituted for the y dimension) is the hypotenuse and it the longest path that experiences the least proper time. Saying the path with the shortest proper time interval is the path that experiences the least proper time is a circular definition. Defining the longest spacetime path length (with time scaled to an equal footing with space) as the path with the least elapsed proper time works in any number of dimensions. For example if we have a clock on the perimeter of a rotating disk then this clock moves in two space dimensions and one time dimension and traces out a spiral path in the inertial frame that remains at rest with the centre of the disk. This spiral path is longer than the vertical path of a clock that remains at rest in the inertial frame and so the clock on the disk experiences less elapsed proper time. In the classic twins experiment, the twin that travels away from Earth and then returns has the longest spacetime path and experiences the least proper time. When the twins experiment is transformed to the point of view of any inertial observer, it always remains true that the twin with least elapsed proper time travels the longest path through spacetime (in SR).

This seems confusing. You are using two different metrics, with different signatures, with the same coordinates and making comparisons between them. Why not 3 different metrics with 3 different signatures?

I also think the statement is false anyway. Consider the path (using (t,x)) from (0,0) to (2,.1) to (0,1). This is 'longer' than from (0,0) to (0,1) using your alternate metric, and *also* longer using the Minkowski metric.

 

[Mentz114]

This seems confusing. You are using two different metrics, with different signatures, with the same coordinates and making comparisons between them. Why not 3 different metrics with 3 different signatures?

I also think the statement is false anyway. Consider the path (using (t,x)) from (0,0) to (2,.1) to (0,1). This is 'longer' than from (0,0) to (0,1) using your alternate metric, and *also* longer using the Minkowski metric.

(In response to yuiop's post #56)

I must say I find this remark baffling. I can't see any of this is the quoted text.

 

[PAllen]

I must say I find this remark baffling. I can't see any of this is the quoted text.

For example:

"It is by this definition that I mean that the longest path through spacetime (with time substituted for the y dimension) is the hypotenuse and it the longest path that experiences the least proper time. Saying the path with the shortest proper time interval is the path that experiences the least proper time is a circular definition."

When he says 'longest' he is using a Euclidean metric; when he says proper time, he is using the Minkowski metric. I find it confusing to use two metrics at the same time.

Then, I gave a specific path such that the longer Euclidean path was also the longer proper time, contradicting his statement.

 

[Mike_Fontenot]

No one in this forum can adequately answer the question because no one has a scientific definition of the word "real".

Einstein formulated and made productive use of a specific definition of "real" in his EPR paper. I did the same in my paper. Our two definitions were different, but they both served a useful purpose.

Mike Fontenot

 

[yuiop]

I also think the statement is false anyway. Consider the path (using (t,x)) from (0,0) to (2,.1) to (0,1). This is 'longer' than from (0,0) to (0,1) using your alternate metric, and *also* longer using the Minkowski metric.

Did you mean a clock traveling (0,0) to (2,0.1) to (0,1)?
The path from (2,0.1) to (0,1) or even (2,1) to (0,1) is unrealistic as it implies a particle traveling backwards in time using (t,x) notation. I perhaps should have added that all particles are restricted to traveling at the speed of light or less but that usually goes without saying. Could you clarify your intention?

 

[PAllen]

Did you mean a clock traveling (0,0) to (2,0.1) to (0,1)?
The path from (2,0.1) to (0,1) or even (2,1) to (0,1) is unrealistic as it implies a particle traveling backwards in time using (t,x) notation. I perhaps should have added that all particles are restricted to traveling at the speed of light or less but that usually goes without saying. Could you clarify your intention?

I was talking geometry, not particle paths. Also getting at the feature that with Euclidean metric you can make strong global statements about extremal properties of geodesics. With SR or GR you cannot. Extremal properties need qualification and with GR are only local at best. Note, geometrically, is there anything wrong with going forward and backward in 'x' along a path? If you are choosing to introduce a Euclidean metric, why should going forward and backward in 't' be any different?

When one discusses Euclidean space, one uses one metric, and pure and simple. You seemed to say using one metric in SR leads to circular definitions. I find this a strange statement which I disagree with.

 

[yuiop]

I was talking geometry, not particle paths. Also getting at the feature that with Euclidean metric you can make strong global statements about extremal properties of geodesics. With SR or GR you cannot. Extremal properties need qualification and with GR are only local at best. Note, geometrically, is there anything wrong with going forward and backward in 'x' along a path? If you are choosing to introduce a Euclidean metric, why should going forward and backward in 't' be any different?

When one discusses Euclidean space, one uses one metric, and pure and simple. You seemed to say using one metric in SR leads to circular definitions. I find this a strange statement which I disagree with.

I agree if we allow particles to exceed the speed of light or travel backwards in time, then yes you can construct "longer paths" through space-time with greater elapsed proper times, but if we restrict ourselves to measurements or real particles then this problem does not occur. Perhaps another of expressing what I was getting at, is that if (and only if) we have two clocks that are initially and finally co-located, then the the clock that has traveled the furthest distance through coordinate space will have accumulated the least elapsed proper time.

In the context of the twins paradox we might explain it like this. Calculate the proper time that elapses for each segment of the each twins journey. Add up the elapsed proper times of the segments. The twin that has the least total of proper times, will as a general rule, be the twin that has experienced the least proper time. This is a true and correct statement, but it is not very helpful to anyone. It amounts to this. Question: Which twin experiences the least proper time? Answer: The twin that experiences the least proper time.

Perhaps the problem lies in semantics. The Minkowski norm of the path is the invariant proper time interval, while I am talking about the Euclidean norm where time is on the same footing (with the same sign) as space. The path with the longest Euclidean norm has the shortest Minkowski norm for paths constrained to the future light cone. Identifying the path with the shortest proper time on a spacetime diagram, can be done at a glance without any calculations using this method and *might* be a helpful aid to beginners. If you think this confusing to beginners then I respect your right to your opinion.

 

[PAllen]

I agree if we allow particles to exceed the speed of light or travel backwards in time, then yes you can construct "longer paths" through space-time with greater elapsed proper times, but if we restrict ourselves to measurements or real particles then this problem does not occur. Perhaps another of expressing what I was getting at, is that if (and only if) we have two clocks that are initially and finally co-located, then the the clock that has traveled the furthest distance through coordinate space will have accumulated the least elapsed proper time.

In the context of the twins paradox we might explain it like this. Calculate the proper time that elapses for each segment of the each twins journey. Add up the elapsed proper times of the segments. The twin that has the least total of proper times, will as a general rule, be the twin that has experienced the least proper time. This is a true and correct statement, but it is not very helpful to anyone. It amounts to this. Question: Which twin experiences the least proper time? Answer: The twin that experiences the least proper time.

If it helps your thinking to have two metrics, so be it. I don't see either the necessity or physical validity of it. When you say: "then the the clock that has traveled the furthest distance through coordinate space will have accumulated the least elapsed proper time" you are introducing a Euclidean metric in an SR problem; for me, this is physically invalid.

 

[Mentz114]

For example:

"It is by this definition that I mean that the longest path through spacetime (with time substituted for the y dimension) is the hypotenuse and it the longest path that experiences the least proper time. Saying the path with the shortest proper time interval is the path that experiences the least proper time is a circular definition."

When he says 'longest' he is using a Euclidean metric; when he says proper time, he is using the Minkowski metric. I find it confusing to use two metrics at the same time.

Then, I gave a specific path such that the longer Euclidean path was also the longer proper time, contradicting his statement.

OK, I missed that somehow.

 

[Dale]

Einstein formulated and made productive use of a specific definition of "real" in his EPR paper.

He actually avoided making a definition of "real", but he did give a definition that he considered a sufficient condition. That was probably a wise choice.

 

[GrayGhost]

yuiop,

I understand your contention. The issue as to whether s is "the longest vs shortest path thru spacetime" is an old one, one that most folks debate at some point or another. This is not an easy topic to convey if addressed properly, so please allow me just a wee bit of time to respond in a good, short, and concise way that lays this to rest ... ie. do it right and do it justice. Hopefully, that'll be sometime this evening.

GrayGhost

 

[GrayGhost]

yuiop,

OK then. Let’s begin as you did, with a standard spacetime diagram and the use of Pathagorus’ theorem, where ict’ is the worldline vector of the moving observer (and hence becomes the hypthenuse) …

(ict’)² = (ict)² + (x² + y² + z²)

Minkowski designates ict’ as the spacetime interval, and so ict’ = s, where s is the length the moving observer travels thru the 4-space (or spacetime) between some 2 events, and he resides at both events …

s² = (ict)² + (x² + y² + z²)

OK, so here’s the big question … Which spacetime pathlength is longer, s or ict? By pathagorean’s theorem, it looks fairly simple … the longer path is the hypothenuse, which is s. In euclidean space, the euclidean metric is used. However as PAllen stated, the metric here is not euclidean, it’s Minikowski’s.

To ask “which pathlength is the longer”, is to ask this … is s < ict? It’s pretty much that simple. We know what ict is, and we know what vt is, and so we need only calculate s.

s² = (ict)² + (x² + y² + z²)
s² = -(ct)² + (x² + y² + z²)
s² = -(ct)² + (vt)²
s² = (vt)² - (ct)²
s² = t² (v²-c²)
s² = c²t² (v²/c²-1)
s = ct (v²/c²-1)^1/2 ... and since i²=-1
s = ict (1-v²/c²)^1/2

which requires that the length of the spacetime interval s be a Lorentz contracted ict, were s = ict’. Therefore, s < ict. Bottom line, the shorter pathlength thru the continuum is s, even though it’s the hypothenuse on a standard Minkowski spacetime diagram.

It’s the use of complex spatial axes (for time) that changes the metric from Euclidean to Minkowski’s. It has to do with the 90 degree counterclockwise rotation of a vector when multiplied by i, or a 180 deg rotation when multiplied by i².

In fact, consider the original formula for the spacetime interval …

s² = (ict)² + (x² + y² + z²)

or, since s = ict’ …

(ict’)² = (ict)² + (x² + y² + z²)

Time must always be orthogonal to space. Yet, there are 2 time axes here, each represented as a complex spatial axis. That is to say, we have 2 axes that are each orthogonal wrt space, “in the same triangle”. So as you can see, there’s more to this equation than casually meets the eye.

That all said, it would be improper in the case of relativity to say that he who travels the longer worldline path experiences the lesser time. Distance thru spacetime is nothing more than the accrued proper-time over a defined interval as reference, and the rate of proper-time is the very same for each and all. He who resides at both events always travels the shorter path, and thus ages the least. If both observers reside at both events, then the one who travels the shorter path ages the least.

Do you agree?

GrayGhost

 

[GrayGhost]

Mike Fontenot,

Wrt your brother/sister acceleration scenario data in ...

http://home.comcast.net/~mlfasf/"​

I haven't verified your data per se, however from basic concepts wrt an accelerated POV, I don't see anything that rubs me the wrong way. Clearly, periods of acceleration and/or deceleration can produce wild clock rates of distant inertial clocks per the borther's POV, so the way the sister's age bounces around do not really look suspicious or anything. And, the inertial periods were fine.

The only data point that I wondered about, just shooting from the cuff, was when the traveler became 26 and his sister 17 ... which occurs on his return leg after completing his +1g burn period away from Earth again. That said, my question is this ... (if you have your data at hand), what are the ages of both at the initial turnabout point when he drops momentarily back into the Earth frame, and (if you know) what was their separation then?

GrayGhost