A question on a tweek of the Twin Paradox

[Zeno Marx]

now the resolution of the twin paradox as put down in textbooks i have read relies on one observer being accelerated and one not so one can claim to be inertial and thus priveledged - this i always felt to be a cheat - even edward teller uses it in conversations on the dark secrets of physics - but what if you had the two twins both accelerated but at different times one accelrated then de-accelerated then brought back to Earth while the other one travels a lot longer at near the speed of light - i think the twin who travels the much longer round trip will still age less though they have both experienced the same accelerations - if you think about this this implies some kind of notion of absolute velocity unless you disagree with the conclusion the one who traveled further at near c will be younger in this scenario?

 

[Mentz114]

There is a simple formula to calculate the ageing along a 4-dimensional path in spacetime. So if we know the details of the trips ( an exact itinerary ) we can calculate what their clocks will read. The ad hoc rule you suggest won't always hold, but the proper time calculation always does.

This article might help

http://en.wikipedia.org/wiki/Proper_time

 

[tom.stoer]

Let's do some math.

Assume we have two twins located at (t,x) = (0,0) in one specific coordinate system. They will meet again at a later time T but at the same location x=0, i.e. at (T,0). The question now is "what are T and T' prime in which coordinate system?".

Now let's avoid coordinates.

Assume one twin is traveling along a curve C from point A to point B in spacetime. The second twin is traveling along a different curve C' from point A to point B in spacetime. Of course we could introduce the coordinates for A and B, but that is not necessary.

Now you have to believe me that the proper time tau of a twin along his curve between A and B is given by the "length" of the curve through spacetime.

$$\tau = \int_C d\tau$$

Here the "length" and therefore the proper time is calculated according to the strange 4-dim. relativistic Pythagoras t² - x².

As the two curves C and C' through spacetime are different for the two twins their proper times will differ.

$$\Delta\tau_{A\to B} = \int_{C_{A\to B}} d\tau - \int_{C^\prime_{A\to B}} d\tau$$

Edit:

Introducing the above mentioned coordinates (t,x) and the velocity v expressed as v = dx/dt we find for the proper time

$$\tau = \int_C d\tau = \int_0^T dt \sqrt{1-v^2}$$

where v=v(t) can be time-dependent. So for two twins starting at t=0 at A and meeting again at t=T at B their proper times and thefore the time dilation as difference of their proper times can be calculated using a difference of these two integrals. Note that both integrals are expressed in terms of a third coordinate time t which is neither identified with the proper time of the first nor of the second twin. In principle and for a physical definition this coordinate time t is not required; the proper times measured with co-moving clocks are sufficient.

This is the most general formula for defining differences of proper time in SRT. It can be used for arbitrary (timelike) curves C with arbitrary speed v.

 

[Dale]

this i always felt to be a cheat

Why? The definition of inertial is pretty clear, as is the fact that the first postulate refers to the equivalence of inertial frames. Everything is up-front and clear, so in what way is it a "cheat"?

but what if you had the two twins both accelerated but at different times

The key point of the twin paradox is asymmetry. A sloppy reading of the first postulate may lead a student to erroneously believe that all motion is equivalent, so the two twins are symmetric. This is the source of the paradox. Once it is pointed out that the first postulate only refers to inertial frames, then it becomes clear that the twins are asymmetric. In your "tweek"ed twin paradox the same resolution applies. They have different accelerations and therefore are asymmetric.

if you think about this this implies some kind of notion of absolute velocity unless you disagree with the conclusion the one who traveled further at near c will be younger in this scenario?

Nonsense. I have no idea how you could even begin to think that is a correct conclusion

 

[Zeno Marx]

Thanks all for your contributions

I was thinking about notions of absolute velocity because I had been reading Lee Smolin's new book "Time Reborn" and he and his mates at the Perimeter institute seem to be proposing a notion of absolute rest which is an observer who will see the CMB the same temperature in all directions so that got me thinking about this

 

[WannabeNewton]

The CMB frame is a privileged frame but it does not define a notion of absolute rest in the sense in which you are describing in post #1.

 

[Zeno Marx]

I still think the proper time integrals unless I have done them wrong support my original naive conclusion though - ie: we have a ceratin "velocity budget" we can spend in either the time or space dimension so the faster we go in space the slower we go in time but this implies some refernce to how fast we are going my point of the tweek to this thought experiment was to remove the preferred observer so the time dilation only depends on your vel;ocity compare d to c over some distance and the observer on Earth can;t claim to be inertial - this is kind of a subtle point but the very form of the proper time integral seems to imply knowing how fast you are going but how can you EVER know what your velocity is if all velocity is relative

 

[Nugatory]

Thanks all for your contributions

I was thinking about notions of absolute velocity because I had been reading Lee Smolin's new book "Time Reborn" and he and his mates at the Perimeter institute seem to be proposing a notion of absolute rest which is an observer who will see the CMB the same temperature in all directions so that got me thinking about this

If you really want to dig into exactly what the CMB does and does not tell you, you might want to ask around the Cosmology forum - doesn't have much to do with twin paradox, whether tweaked or not. Here's one thread from there.

 

[Zeno Marx]

you have to put a figure in for 'v' but where do you get this from?

 

[WannabeNewton]

Are you asking where the $v$ comes from? $\int d\tau = \int \sqrt{dt^{2} - dx^{2}} = \int dt\sqrt{1 - (\frac{\mathrm{d} x}{\mathrm{d} t})^{2}} = \int dt\sqrt{1 - v^{2}}$.

 

[Zeno Marx]

i tseems we can only ever calculate velocity for others rather than ourselves but accelerations and gravitational fields sem to slow and speed clocks in a way which is totally path dependant so if you get accelerated to a certain speed your clock locks at that rate and then satys there even if your frame is then inertial - infact i have a wacko suspicion that inertia is actually the resistance to change in the internal time metric the same way pressure and tension are resisatnce to the change in the internal spatial metric but hell that's just me playing with equations and drawing weird conclusions

 

[WannabeNewton]

We can certainly calculate our own velocity: in our frames our velocity is zero. The rest of your post is nonsensical I'm afraid.

 

[Zeno Marx]

Are you asking where the $v$ comes from? $\int d\tau = \int \sqrt{dt^{2} - dx^{2}} = \int dt\sqrt{1 - (\frac{\mathrm{d} x}{\mathrm{d} t})^{2}} = \int dt\sqrt{1 - v^{2}}$.

I still don;t see how you haven;t put this into the equations by hand at some point - relative to what? we can it seems only draw conclusions about the universe relative to us but nothing about ourselves but the fact our clocks are alterered by gravitational fields to me says they are also altered by acceleraqtions and velocity then means something in the same sense gravitational potential does?

 

[Zeno Marx]

I mean change in velocity means something in the same sense as change of gravitational potential as far as how fast your clocks go

 

[WannabeNewton]

Yes we can relate the observed differences in clock rates due to acceleration to the same result in stationary gravitational fields via the equivalence principle:
https://www.physicsforums.com/showthread.php?t=310019
and see here for something related:
http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_gr.html

 

[Nugatory]

you have to put a figure in for 'v' but where do you get this from?

You don't have to start with a known v. You can start with the known position of the object in space-time (its x and t coordinates) as a function of any convenient parameter, and then do the integration with respect to that parameter. It may be convenient to use the x and t values from a frame corresponding to an observer moving at a particular speed, but you can choose any speed you please; the x and t functions will take different forms but the integral will come out the same.

 

[pervect]

Newtonian and relativistic physics can both be written in terms of the principle of least action. I.e, there is some quantity , the action which is minimized (or more exactly locally minimized or extermized) by the natural motion of a body.

It turns out that the "action" of a point mass is just - m * proper time. This means that the equations of physics themselves say that a body that moves naturally (i.e. isn't subject to any external forces to make it acclerate) maximizes proper time.

There are several subtle points that I"ve glossed over, such as the difference between "extremal", "maximal" and "minimal", but these are the basics behind the "twin paradox". Learning a bit about Lagrangian physics and the principle of least action will be much, much, much more productive than speculations in the direction of some "absolute velocity".

 

[Zeno Marx]

Newtonian and relativistic physics can both be written in terms of the principle of least action. I.e, there is some quantity , the action which is minimized (or more exactly locally minimized or extermized) by the natural motion of a body.

It turns out that the "action" of a point mass is just - m * proper time. This means that the equations of physics themselves say that a body that moves naturally (i.e. isn't subject to any external forces to make it acclerate) maximizes proper time.

There are several subtle points that I"ve glossed over, such as the difference between "extremal", "maximal" and "minimal", but these are the basics behind the "twin paradox". Learning a bit about Lagrangian physics and the principle of least action will be much, much, much more productive than speculations in the direction of some "absolute velocity".

I'm very much simpatico with least action physics richard feynman being my major scientific hero my point was really - ok maybe the thought experiment i posted didn;t do it properly but the idea wa to ask how you define velocity at all if you eliminate totally the concept of inertial observers because no observer over the course of his/her/its observership is ever completely inertial

 

[Dale]

how you define velocity at all if you eliminate totally the concept of inertial observers because no observer over the course of his/her/its observership is ever completely inertial

Even if an observer is non-inertial you can still have an inertial reference frame. There is no need to use a frame where a particular observer is at rest, you can always use any inertial frame.

I.e., despite the traditional shorthand way of speaking, observers and reference frames are not the same thing.

 

[Zeno Marx]

but my problem is how you define velocity WITHOUT inertial frames because we are all experiencing accelerations all the time so there is really not such a thing as an inertial frame except in deep space

 

[Dale]

but my problem is how you define velocity WITHOUT inertial frames

There is no need to do this.

we are all experiencing accelerations all the time so there is really not such a thing as an inertial frame

You completely missed my point above. Please re read it. A reference frame is a mathematical construct and is in no way dependent on whether or not real objects experience acceleration. Even if EVERYTHING is accelerating you can still define an inertial frame. All that will happen is that nothing will stay at rest wrt the inertial frame.

 

[pervect]

I'm very much simpatico with least action physics richard feynman being my major scientific hero my point was really - ok maybe the thought experiment i posted didn;t do it properly but the idea wa to ask how you define velocity at all if you eliminate totally the concept of inertial observers because no observer over the course of his/her/its observership is ever completely inertial

If Newton's laws hold well enough that you can use them, then you have a de-facto inertial observer. This is most of the time. You can then describe things in the famliar manner of inertial frames, and inertial observers, etc, just by going to any of the various coordinate systems in which Newtons laws work well, or well enough that you can't tell the difference.

Occasionally, one finds situations (such as considering the universe on a cosmological scale) where one can't approximate physics via Newtonian laws.

If curvature is present (and in a truly general situation I would expect it to be - no curavature is a special case) I would say that there isn't any good , truly general, coordinate independent definition of velocity except for two observers who are close to each other. See for instance Baez's remarks in "The Meaning of Einsteins' Equations."

http://math.ucr.edu/home/baez/einstein/node2.html

In general relativity, we cannot even talk about relative velocities, except for two particles at the same point of spacetime -- that is, at the same place at the same instant. The reason is that in general relativity, we take very seriously the notion that a vector is a little arrow sitting at a particular point in spacetime. To compare vectors at different points of spacetime, we must carry one over to the other. The process of carrying a vector along a path without turning or stretching it is called `parallel transport'. When spacetime is curved, the result of parallel transport from one point to another depends on the path taken! In fact, this is the very definition of what it means for spacetime to be curved. Thus it is ambiguous to ask whether two particles have the same velocity vector unless they are at the same point of spacetime.

There might be some ways to avoid some of the impact of what Baez says in specific cases - for instance, taking advantage of particular symmetries, such as the "Hubble Flow" in the case of the universe at large.

My experience is that a lot of physicists do want to define relative velocity, and they use various tricks to try to avoid the issues Baez mentions, with various degrees of success. So the idea of velocity can be useful, but one needs to pay careful attention to how it's being defined in any specific case - it's usually based on some trick or feature of the particular problem at hand or the use of some particular coordinates.

Semi-philosophically, though, there isn't any need for relative velocity - and in fact, we don't even really need observers, either.

Consider Misner's remarks in http://arxiv.org/abs/gr-qc/9508043 "A Precis of General Relativity". It was written in response to some remarks by Neil Ashby,.

A method for making sure that the relativity effects are specified correctly (according to Einstein’s General Relativity) can be described rather briefly. It agrees with Ashby’s approach but omits all discussion of h ow, historically or logically, this viewpoint was developed. It also omits all the detailed calculations. It is merely a statement of principles.One first banishes the idea of an “observer”. This idea aided Einstein in building special relativity but it is confusing and ambiguous in general relativity. Instead one divides the theoretical landscape
into two categories. One category is the mathematical/conceptual model of whatever is happening that merits our attention. The other category is measuring instruments
and the data tables they provide.

For GPS the measuring instruments can be taken to be either ideal SI atomic clocks in trajectories determined by known forces, or else electromagnetic signals describing the state of the clock that radiates the signal.

<...snip...>

What is the conceptual model? It is built from Einstein’s Gen
eral Relativity which asserts that spacetime is curved. This means that there is no
precise intuitive significance for time and position. [Think of a Caesarian
general hoping to locate an outpost. Would he understand that 600 miles
North of Rome and 600 miles West could be a different spot depending on
whether one measured North before West or visa versa?] But one can draw
a spacetime map and give unambiguous interpretations.

Misner goes on to explain that the "space-time map" is just a metric - a mathematical construct which allows one to calculate geodesics ("straight lines" - which can include the motion of force-free bodies as disucssed above) and their lengths (which for moving bodies, are time-like, the proper times we've been discussing).

So really, the only reason we need to talk about 'observers' is to try to allow people to leverage as much as they can off their Newtonian intuition. They aren't actually needed for anything, and frequently there is a lot of confusion generated by trying to pound the square peg of the mathematics of General Relativity into the round hole of Newtonian-based preconceptions.

To expand on this viewpoint a bit, in General Relativity, we can assign labels to events in space-time in any manner that we find convenient, and from any such assignment, we can create a metric, a "map" of space-time, which assigns particular labels (which we call coordinates) to specific events.

These labes are purely human inventions - though typically, some will be easier to work with than others, when they respect fundamental physical symmetries, for instance. But the point is that the coordinates are not "physical", they're human inventions. A very simple point, but it seems difficult to get people to appreciate it :-(.

A useful replacement for "an observer" using this semi-philosophical approach is to define some particularly useful coordinates called Fermi normal coordinates. Given any particular worldline, such coordinates can be constructed, and will behave in a manner that is, in some sense, as close to Newtonian as one can manage.

I've seen a few people on PF who seem to prefer a different set of coordinates, which are based on a particular point in space-time, rather than a particular worldine. These can be useful, they don't to me have the same feature of being associated with an "observer", because they are associated with only one point in space-time, not a worldline that moves "through" it.

I'm not sure it'd be relevant to get into more details of Fermi normal coordinates, and this has gone on more than long enough already, so I'll cut it here.

To recap, though - we don't really NEED to define velocities - or even observers - to do physics. We can make all the physical predictions we need from the metric.

 

[swampwiz]

now the resolution of the twin paradox as put down in textbooks i have read relies on one observer being accelerated and one not so one can claim to be inertial and thus priveledged - this i always felt to be a cheat [sic] ...

I was having the same difficulty with this concept - but then I recalled that a moving reference frame is length contracted, and if one of the twins is going to visit a distant star, then at the exact moment (i.e., in case the star itself is moving relative to the Sun, and as observed by the reference frame of the Sun) that twin visits the distant star, essentially there is an astronomically long yardstick between where that star is at that exact moment and the Sun, and which is in the frame of reference of the Sun, and therefore is observed to be length contracted in the frame of the traveling twin. The relative velocity with respect to the Sun, however, is not length contracted, since it is always the same (although in opposite directions) for either reference frame.

So the net result is that as far as the traveling twin is concerned, the distance he observes as traveling is less than the distance the Earthbound twin observes, while both observe the relative velocity of the traveling twin as the same.