Yet Another Run at the Twin “Paradox”

What’s with the “if” in this question? And what do you mean by the word “co-moving” other than perhaps stationary with respect to one another?
The standard version of the Twin “Paradox” starts with two twins, who are obviously “co-moving” in that sense at birth, and a distant star which is also “co-moving” (ie stationary with respect to the twins). Then at some point one of the twins travels to the star and back (usually with unspecified periods of acceleration and mostly constant speed in both directions).
Any correct application of Special Relativity predicts that when they re-unite the traveller is younger. The age difference can be calculated in terms of any frame of reference and (for any specified acceleration history – including that of instantaneous speed jumps) the answer is always the same so there is no real paradox.
The alleged “paradox” arises only in the mind of someone who notices that the traveller perceives the homie to have been ageing more slowly during the constant-speed legs of the trip and then just ignores the fact that the traveller also perceives a sudden rapid ageing of the homie during the turn-around. (During that turn-around the traveller feels the force of acceleration and so is aware of being in a non-inertial frame, whereas the homie feels no such forces. So the situation is definitely NOT symmetrical.)

Source: (1001) Alan Cooper’s answer to How do you solve the Twin Paradox if everyone (including the waypoint) is co-moving at the get-go? – Quora

Wigner’s Friend

If we define both the observer and the “observed” as both being part of say an even bigger system, would the wave function still collapse in this system?

This conundrum is known as the Wigner’s Friend problem, though it is also often asked with reference to Schrodinger’s cat.

In my opinion its best resolution is in the understanding that the wave function or quantum state is not a property of the system itself but of its relationship to an observer, and I think this view is a better reading of what Hugh Everett was describing in his “Relative State” interpretation of quantum mechanics [which was re-presented later (mostly by others) as a “Many Worlds” interpretation where observations (and other interactions) continually cause the creation of new “branches” (in a way that Everett himself apparently once described as “bullshit” in a marginal note on someone else’s elaboration of the MWI)].

Source: (1001) Alan Cooper’s answer to Is the collapse of the wave function in Quantum Physics based on a system frame of reference? If we define both the observer and the ‘observed’ as both being part of say an even bigger system, would the wave function still collapse in this system? – Quora

Is observation required for collapse? 

Whether or not observation is the only way in which a wave function can collapse depends on what you mean by “collapse”, and that word is used by various people in reference to different aspects of the measurement and observation process – which can be considered as happening in two stages.

The setting involves a system in a pure quantum state which may have been prepared as an eigenstate of some observable (such as spin relative to a particular direction), and so is a nontrivial superposition of eigenstates of some other observable (such as spin relative to a different axis) which we now want to measure.

In the first stage, the system of interest interacts with a larger more complex system which is not fully known and so is in a statistical mixture of pure states (represented by a density matrix rather than a single state vector). If the larger system is suitably designed as a measuring apparatus, then the interaction leads to the state of the combined system approaching a statistical mixture of states in which the subsystem of interest is in an eigenstate of the observable and the measurement apparatus is in a related state which involves some macroscopic feature (such as a pointer, a readout panel, or a bright spot on a phosphor screen) which has a corresponding humanly visible value. Henceforth the system acts as if it is in just one eigenstate which is not yet known but is subject to classical probabilities. This process eliminates the possibility of future interference between the eigenstates that was possible while the state of the system was in a pure state (represented by a coherent wave function) and so is often called “decoherence”; and since it reduces the system to being effectively in just one eigenstate it is often identified with “collapse of the wave function”. It actually happens in almost any interaction with a complex system (even when there is no humanly visible related macroscopic property of the system). So, for those who identify decoherence as collapse, it is indeed possible for collapse to occur without observation.

But after this kind of “collapse” we still don’t know what the measured value actually is, even though we can think of it as having just one of several precise values – each with some known probability.

The second stage of the observation process is where the conscious observer notices which value is present. Some people think of this as where the “collapse” happens, but here it is not really collapse of the wave function but rather of the classical probability distribution (similar to the case of a coin toss which starts of in a stochastically mixed state and collapses to just one case when we see the result).

The difference from a coin toss is that in that case we assume that all along the system was really in whatever particular state we eventually observe, and that state could have been determined with certainty just by making more observations at the start; whereas in the quantum situation the uncertainty seems to be essential until we actually experience the result. This leads to a philosophical problem for those who think that the quantum state is a property of the system itself rather than of its relation to the observer as it seems to imply that the experience of a conscious observer has some physical effect on the universe and raises the problem of Wigner’s friend who watches an experiment before Wigner does and seems to collapse the wave function even though the friend is himself just a complex quantum system who Wigner sees with a wave function that does not collapse until the information reaches his (Wigner’s) own mind.

To my mind this is resolved by seeing the quantum state as a description not of the universe but of its relationship to the observer; and I think this view is a better reading of what Hugh Everett was describing in his “Relative State” interpretation of quantum mechanics which was re-presented later (mostly by others) as a “Many Worlds” interpretation where observations (and other interactions) continually cause the creation of new “branches” (in a way that Everett himself apparently once described as “bullshit” in a marginal note on someone else’s elaboration of the MWI).

Source: (1001) Alan Cooper’s answer to Is observation the only way in which a wave function can collapse? – Quora

Does an observer modify the observed?

What many people misunderstand is that in quantum theories the “state” of a system is not a property of the system itself but rather of how it appears to an observer.

There are actually at least two stages to the observation process. One is when the system of interest interacts with the much more complex system of a measurement apparatus whose precise quantum state is too complex for the observer to keep track of and so has to be expressed as a statistical mixture. This can have the effect of causing the combined system, in which the observed subsystem was initially in a pure “coherent” superposition state (with interference still being possible between different possible observed eigenvalues), to end up very close to a statistical mixture in which each possible measured value of the observed quantity has a well defined value with no interference between them. This “decoherence” process can be caused by interaction with any sufficiently complex system (even, as Viktor Toth notes, just a brick) and it does modify the observed (as does any interaction with anything – even just another simple quantum system). But it still leaves the actual value of the observation unspecified. The “collapse” process, which identifies which particular value has occurred, only happens in the mind of the observer whose conscious experience corresponds to just one of many possible histories of the universe. But this doesn’t modify the observed – at least no more than it modifies everything in the universe that is dependent on that observed value. (For example if we are in a room together and I see a red flash then the you that I see will also see a red flash, but if you see a blue flash then the I that you see will also have seen a blue flash.)

Source: (1001) Alan Cooper’s answer to In the quantum mechanical idea that the observer modifies the observed, can the observer be an insect? – Quora

Acceleration effect on light speed

The idea that the experience of an accelerated observer might be approximated by considering its worldline as comprising many small inertial pieces is a good one. And during each inertial step the speed of light seems to be constant everywhere. But at the velocity boosts or “frame jumps” between the steps, the apparent coordinates of all events (including those on the world line of a light signal) get shifted, so the light seems to jump ahead or back. Taking the limit of these approximations leads to the conclusion that the light signal does not seem to have constant velocity from the point of view of the accelerated observer. (Since the “frame jumps” lead to coordinate changes that are proportional to the distance of the event from the observer, this does not change the fact that every light signal seems to have the same speed when it reaches the observer, so there is no local change and it is just when the signal is far away from the observer that its velocity appears to vary.)

Source: (1001) Alan Cooper’s answer to A rotating frame can be divided into an infinite number of infinitesimal inertial frames. According to SR (special relativity) the light speed in each inertial frame is constant. Is light speed therefore constant in say the Sagnac experiment with SR? – Quora

Perspective on “Moving”

A Quora question asks:
From your own perspective, frame of reference, if you are not accelerating, are you not moving at all?

Perhaps the reason people find this difficult to grasp is because of the ambiguous and unnecessary words “at all” at the end – which lend support to the idea that the word “moving” without mention of a context has any meaning at all. It does not.

And it never has. The claim that something is “moving” has never had any meaning without identification of relative to what, and it is perhaps unfortunate that by convention we often take the word without modification as meaning relative to the Earth without explaining that convention to small children (who then grow up thinking that there must be some kind of absolute property of “moving”).

From my own perspective, while sitting still in a bus I am not moving relative to the bus, but I know damn well that I am moving relative to the world outside. And it’s not just from my own perspective. Everyone else agrees with both of those relative motion statements.

So although it is just a tautology that from every perspective I am never moving relative to myself, it is false (or rather meaningless) that there exists any perspective from which I am ever not moving at all.

Source: (1000) Alan Cooper’s answer to From your own perspective, frame of reference, if you are not accelerating, are you not moving at all? – Quora

Symmetry of “Inertial Mass”

In any elastic collision between high velocity particles, conservation of momentum, expressed in its naive form in terms of the initial frame of either particle, requires that the “inertial” mass used for the other (and for both of the post-collision particles) include the relativistic gamma factor(s). So it is indeed true that twins moving at constant velocities in different directions BOTH infer the other’s inertial mass to be more than their own.

Source: (1000) Alan Cooper’s answer to Ok, Newton’s physics was wrong. But is there any experimental evidence suggesting Einstein’s relativity claim is true that twins moving at constant velocities in different directions BOTH measure the other’s inertial mass to be more than his own? – Quora

What is relative in Relativity?

What is relative in any physical theory of “relativity” are the space-time coordinates of events from the perspectives of different observers. 

One problem, I think, with the names using ‘Theory of Relativity’ is that they seem to suggest theories about what is relative, rather than (more correctly) about how the coordinates used by different observers need to be related in order to ensure that the laws of physics are invariant (ie NOT relative).

In fact the coordinates that seem most natural to any observers for the purpose of expressing their experience in quantitative terms are always to some extent relative to the observers, so just saying that they are relative without specifying how is not telling us much (though in the new theories there is “more” relativity in the sense that time as well as the spatial coordinates becomes relative).

Our intuitively expected relationship between the coordinates of relatively moving observers allows all observers to use the same time coordinate, and so to agree on which events are simultaneous (ie constitute the same moment in time). It also preserves the form of Newton’s equations of motion for observers moving at constant relative velocity – which, as Galileo noted, has the consequence that observers moving with constant relative velocities cannot, by mechanical experiments, identify any particular one as being stationary. So the question of who is moving can only be answered relative to a particular observer – but this is just one particular instance of the relativity of coordinates.

[Sometimes observers moving relative to some larger object such as the Earth might choose to agree on a fixed Origin based on that object rather than on their own positions. But Galileo noted that if they are all moving together inside a moving vessel without any view of the outside, then it makes sense for them to use the vessel itself as their frame of reference – and relative to that, anything outside would appear to be moving in the opposite direction. In the world of Galilean/Newtonian physics there is nothing aside from its greater size which makes us prefer the Earth’s frame to that of the vessel, nor anything besides Earth’s proximity which makes us prefer its frame to that of the Sun. The answer to whether or not anything is or is not actually moving was thus, even in classical mechanics, entirely relative to the observer’s arbitrary choice of a frame of reference; and so that certainly was NOT anything new in Einstein’s theory.]

The above noted preservation of form of the equations of motion is perhaps confusingly called both “Galilean invariance” and “Galilean relativity”. The confusion could be avoided by making it clear that the word “relativity” applies to coordinates and “invariance” to the laws of physics. But I think that the practice of using “relativity” for the invariance itself rather than for the coordinate transformations under which it holds was indeed a misnomer which I believe precedes Einstein (though as an aside I must add that it seems surprisingly difficult to find out who was actually the first to do this).

Einstein’s special theory describes how the spacetime coordinates must be related in order for the laws of electromagnetism to have the same form for all inertial (ie unaccelerated) observers in the absence of any gravitational field. It turns out that for this to work, observers in relative motion will not be able to use the same time coordinates, and indeed will have different notions of simultaneity; so in this theory there is indeed something more that is “relative” than in the Galilean theory (but I don’t think that is why the theory got its name).

Einstein’s theory derives the relativity of simultaneity, and the formulas relating spacetime coordinates of different observers, from the principle of invariance of Maxwell’s equations (and so in particular, invariance of the speed of light) from the points of view of all inertial observers. But in my opinion Einstein’s reference to that principle as the “principle of relativity” (as opposed to the “principle of invariance” as suggested for example by Felix Klein) was indeed a misnomer, and apparently even Einstein eventually expressed some agreement with this  (but too late to actually change it).

[The special theory of relativity also includes modifications of the laws of mechanics (excluding gravity) which are necessary for them to remain invariant under the same transformations as those which preserve Maxwell’s equations – but this has nothing to do with the name except for the fact that perhaps the thinking was that the “principle” in question was that all physical laws need to be invariant under the same relativity of coordinates.]

The general theory goes on beyond the special theory to describe how the coordinates should be related in order to preserve an invariant form for both electromagnetic and gravitational forces under more general conditions (including accelerated observers and gravitational fields). So it’s not that more things are relative in the general theory, but rather that the relativity of the same things is explored under a more general range of conditions.

P.S. It should perhaps be noted that, just as the special theory has no distinguished inertial frame, the general theory does not provide any purely local way to distinguish inertial from accelerated frames as no accelerated observer can distinguish the experience of being accelerated from that of being prevented from falling freely in some “fictitious” gravitational field – which can only be identified as truly fictitious by observing the absence of possible sources (mass-energy distributions) out to an arbitrarily great distance. So there is some sense in which acceleration vs gravitation distinction is not quite absolute in the general theory but I don’t think that this (or the absence of any distinguished inertial frame in the special theory) was ever the reason for our use of the word “relativity”.

Source: I am trying to understand the term general relativity: what is relative in GR? Gravitation (acceleration) is absolute, not relative! So what is relative in GR? Is it perhaps a misnomer? – Quora

Days of Future Past?

Has the future already happened according to special relativity? – NO.

In fact, in special relativity, the question of whether or not an event has “already happened” depends on the observer and has no meaning if the observer is not specified.

I find it so hard to believe!! – THEN DON’T.

Believe this instead (but only after making sure that you understand it):

What is true according to special relativity is that for any distant observer relative to whom you are moving sufficiently rapidly, some events in your future may be seen as in their past relative to the time on their clock at which you think they are now (or rather at which you will think they were now when you eventually see that “now” event in their lives).

[And for every event in your future there are some possible observers in your “now” (though you will not have actually seen them yet) who, when they finally see that event, will judge it to have happened in their past relative to the time on their clock at which you (will) think they are now.]

So in the world of special relativity, there is no time-ordering of events that all observers will agree on.

Source: (151) Alan Cooper’s answer to Has the future already happened according to special relativity? I find it so hard to believe!! – Quora