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Another Twins Answer

Why should the twin on the spaceship be younger than the other on earth if each of them is supposed to observe the time dilation of the other in his own frame?

The question of which is younger when they are apart and in relative motion has no answer unless we specify the observer who is making the comparison (which could be either of them – or perhaps some other arbiter such as one who is stationary with respect to the Cosmic Microwave Background radiation).

Once they reunite they, and everyone else, will agree that the one who ends up younger is the one who experienced more acceleration towards the other when they were far apart (or more precisely for whom the integral of distance times the negative of its second derivative is greatest). But even though they will agree on the end result, they won’t agree on a moment-by-moment accounting of how their ageing rates compared.

Source: (1000) Alan Cooper’s answer to Why should the twin on the spaceship be younger than the other on earth if each of them is supposed to observe the time dilation of the other in his own frame? – Quora

What’s wrong with saying “General relativity says that the orbiting clocks should tick about 45 millionths of a second faster than they would on Earth”?

There are three main errors in that sentence.

One is more a matter of poor wording but still an error, the second is perhaps just a misprint, and the third is I think at the heart of many misunderstandings of Relativity.

First the minor stuff:

What General Relativity predicts for an orbiting satellite can be written as a sum of two parts, one gravitational (which is the same as what GR predicts for a stationary clock at the same height as the satellite) and the other sometimes called kinematic, but that breakdown is only approximate as there are other higher order terms involved as well (which include products of gravitational and velocity factors). What the quoted sentence has called the prediction of GR is just the gravitational part (which does indeed contribute about 45 microseconds to the daily time advance of the satellite clock). But GR also predicts the kinematic part (of about 17 microseconds per day of retardation), as well as those smaller additional terms, and so it is wrong to identify just the gravitational term as what GR predicts rather than as what GR would predict if the satellite were stationary (and NOT “orbiting”). The writer also left off the “per day” in describing the pure gravitational term, but I am sure that was just a misprint.

But, as I said, all that, though definitely wrong, is relatively minor stuff.

The more serious error is in the description of having a daily advance (or in the purely kinematical part a daily retardation) as ticking “faster” (or slower). It would be perfectly ok if you said ticking faster (or slower) on average, and leaving off the “on average” might seem like a small thing; but it is a serious error because it leads to the wrong idea that it means ticking faster (or slower) all the time. And the reason that is wrong is because different observers (while agreeing on the accumulated time differences recorded on two clocks between times when they are together) may disagree on whether one was always running slower or sometimes slower and sometimes faster (by amounts that give the same net total as in the always slower case). And we have no way of deciding which of them is “right”.

Source: (1000) In the twin paradox it is often stated that the clocks can only be compared at the same location. Why can’t the clocks be compared at space stations synchronized with the earth clock on the travelling twin’s journey? – Philosophy of Relativity – Quora

Is it wrong to assert that a GPS satellite clock runs slower due to kinematical time dilation?

Yes, if you want to be precise about it, and for at least two reasons.

The GPS satellite clock will actually appear to run faster (on average) for all observers (and faster at every time for both itself an observer standing on the Earth, but possibly sometimes slower for an observer in free fall down a deep well). I know you referred to “kinematical” time dilation but that isn’t really a separate thing as the separation of effects into “kinematical” and “gravitational” is only an approximation to the full GR effect.

But it’s still wrong if we just define the “kinematical” part of the approximate split as the effect that would be seen if the satellite was not in orbit but just moving at constant speed in a straight line. In this case it is true that what was the satellite clock now appears from Earth to be slow; but it also appears to the (ex)satellite observer that it is running faster than the Earth clock. And unless we specify a way to decide who’s impression is right then it is wrong to say that either clock actually is running slower.

In the twin paradox it is often stated that the clocks can only be compared at the same location. Why can’t the clocks be compared at space stations synchronized with the earth clock on the travelling twin’s journey?

The traveller’s clock can indeed be unambiguously compared with each space station clock at the event where they pass by one another, but that is still only comparing clocks when they are at the same location. And the problem with saying that comparing one’s time with that on a space station is equivalent to comparing it with the one on Earth is that it depends on agreeing that the space station clocks are properly synchronized. But if the space station clocks appear synchronized with the Earth clock in its own frame, then they will not appear synchronized to the traveller who is passing by them. So the time on the space station clock does not match the traveller’s idea of what is the current time back on Earth.
One can indeed go through the process of keeping track of the space-station clock times compared to the traveller’s clock, and will find that those recorded times are all greater on the space-station clocks by the same Lorentz gamma factor. But so long as the velocity remains constant, the traveller could be part of a lined up fleet of ships all moving at the same velocity past the Earth (and so stationary with respect to one another with the Earth and space stations moving past them), and if they all synchronize their clocks with the traveller then the Earth and space station clocks will record the intervals between successive ships of the fleet as greater than the time differences between the clocks on those ships. In other words the Earth (and space station) observers see the ship times as more closely spaced than their own and the traveller (and fleet ship) observers see the times on space station clocks as more closely spaced than the times (on their own ship-based clocks) at which they pass by them. At first sight perhaps this looks like a paradox, but we need to note that each observer of either kind is comparing times on different clocks of the other kind with successive times on the same clock of their own and each can attribute the effect to an assumption that the other set of clocks is not properly synchronized. So this isn’t really a paradox, but there is still no way of deciding which team is actually synchronized and which is not – and without being sure of that the traveller can’t rely on the space stations as true representatives of the time back on Earth.
Making the traveller turn around and return to Earth is just one way of getting some particular pair of clocks back together for an unambiguous comparison of time intervals. (Another would be to have the Earth chase after the traveller and compare notes when she catches up, and yet another would be to do things symmetrically.) But they all involve having someone change their inertial frame (ie accelerate) and the result depends on the acceleration pattern but is always basically that the one who experienced the most acceleration towards the other when they were far apart is the one who will end up younger.

[In the symmetrical twins story both end up the same age, and are not surprised because each has seen the other age first more slowly and then more rapidly but ending up with exactly the same total amount of ageing as they themselves have experienced. If they use the light travel time to infer when each tick of the other’s clock actually occurred (as opposed to when they see it), then each will infer that the other’s clock was running more slowly during both constant speed parts of the trip, but more rapidly during the period when they felt the force of acceleration during the turn-around process – with the same final result.]

Source: (1000) Alan Cooper’s answer to In the twin paradox it is often stated that the clocks can only be compared at the same location. Why can’t the clocks be compared at space stations synchronized with the earth clock on the travelling twin’s journey? – Quora

Whose idea of Simultaneity is Wrong?

Two inertial observers in motion relative to one another may disagree as to which of two events happened before the other.

They are both drawing reasonably sensible and natural conclusions from their respective observations – assigning what they actually see to a time that is earlier by the light travel time (which they compute from the observed distance and the measured speed of light). But since they disagree one or both must be wrong if they both think their own idea of simultaneity is the only “right” one.

It could be that one of them (or perhaps some other inertial frame) is “right” in some sense and all the rest are “wrong”. But we don’t have any way of exactly identifying the “right” frame, and using that fact as a starting point provides a way of making accurate predictions without needing to make the untestable assumption that some arbitrarily chosen particular frame is in fact the special one.

Source: (1000) Alan Cooper’s answer to In twin paradox, the traveller’s clock ends up with a lesser total elapsed time, so we can tell who made the trip. Does this not contradict the postulate of SR that all physical laws are the same in all frames and all inertial frames are equivalent? – Quora

Where in the universe can we find such an inertial frame? Certainly not on the surface of earth!

SR only applies exactly in the absence of gravity. So in the real world it is just an approximation that works well enough for predicting things where the effect of gravity is small (such as interactions between small high velocity particles in accelerators near the Earth’s surface, or between spacecraft and small bodies like asteroids far from planets, but not for things like apples falling out of trees on Earth).
In regions where it does provide a good approximation, it works just as well for accelerated as unaccelerated frames, but for accelerated frames the formulas needed to express physical laws in terms of the observer’s coordinates are more complicated.

So the excuse used NOT to apply relativity theory in the twin paradox is a brief period of zero seconds at the turnaround point?

No one who knows what they are talking about has suggested “NOT to apply relativity theory”. On the contrary, the correct application of relativity theory leads to the conclusion that when the twins re-unite they agree on the fact that they have both seen the traveller age less. They just disagree on when during the trip the Earth-based twin aged faster. The one on Earth thinks it happened at a steady rate throughout the trip and the traveller (after actually seeing it during the return trip) thinks (after making the light travel time correction) that it happened quickly during the turn-around.

Prior to the turn around, each sees the other ageing more slowly (due to the Doppler effect) and, even after making the light travel time correction, thinks that part of that slowdown remains unexplained (and so in some sense is “really” happening).

But any claim that during the outbound journey “we know for a fact that the travelling twin is younger than the earth twin” (or vice versa) is completely false. There is nothing that is absolutely true about the relative ages of the twins until they are at rest with respect to one another.

More on “Collapse”

The “state” of a system that is studied in quantum mechanics is not a property just of the system itself, but rather it is a summary of what is “known” about that system by a class of outside “observers” who do not interact with it in any way between “observations”. (The “observers” don’t actually have to be conscious; anything such as a measurement apparatus that could be changed in some macroscopic way by interaction with the system could play the same role, with the relevant change being identified as “knowledge” so long as different macroscopic states of the “observer” end up associated with different values of whatever property is being “observed”.)

This relative state changes whenever the observer learns something about the system – and when that happens the probabilities of all values other than the one observed become zero (while the value experienced by the observer becomes certain from the observer’s point of view). This change is sometimes called “collapse” – though it should be noted that the total of all the probabilities remains the same, so it might be better to think of the probability distribution being collapsed “sideways” onto the observed value rather than “down” to zero everywhere.

As an example, consider the case of an electron that has just passed through a selector that ensures that its spin in the z-direction is positive (let’s call this “up”). If we think of the spin of the electron as a system in its own right and if we can control the path of the electron without interacting with its spin, then we can represent (what we know about) the spin by a quantum state, and it turns out that if we know the spin is up then we cannot assign any particular value (say left or right) to any horizontal component. All we can say is that if we pass many such electrons through another vertical spin selector they will all measure spin up, but if we measure in any horizontal direction we’ll get positive and negative results with probability 1/2 each. BUT (and here’s the “collapse”) if, on doing that horizontal measurement, we observe spin (say) left, then when we subsequently measure the same horizontal spin we’ll get the same result with certainty and the probability of seeing right will have gone down to zero (while if we go back to measuring the vertical component it will now have equal chances of being up or down – which is another reason for not liking the “collapse” language since what collapses one distribution has spread out another, and in particular has raised the chance of seeing down next time from 0 to 1/2).

Note: Jonathan Joss makes a similar objection to the word “collapse” and suggests that it be called a “rotation” of the state vector (but of course in the relevant Hilbert space rather than physical space).

Source: (1001) Alan Cooper’s answer to Can you give some examples where an object’s state changes due to its being observed by an outside observer (observation collapsing wave functions)? – Quora

Dead or Alive?

All I know for sure is that any cat whose owner would subject it to such an experiment cannot have been “Wanted”.

But if Schrodinger had had many cats to try this on I guess about half of them would be dead and the other half alive. And in all cases the “collapse” would have occurred long before the box was opened – or rather never, because a cat could never be put into a pure quantum state in the first place.

Source: (1001) Alan Cooper’s answer to After Schrodinger opened the box with the cat inside, and thoroughly observed the cat, collapsing the superposition, was it dead or alive? – Quora

(1001) Alan Cooper’s answer to Can you give some examples where an object’s state changes due to its being observed by an outside observer (observation collapsing wave functions)? – Quora

The “state” of a system that is studied in quantum mechanics is not a property just of the system itself but rather is just a summary of what is “known” about it by a class of outside “observers” which do not interact with it in any way between “observations”. (The “observers” don’t actually have to be conscious; anything such as a measurement apparatus that could be changed in some macroscopic way by interaction with the system could play the same role, with the relevant change being identified as “knowledge” so long as different macroscopic states of the “observer” end up associated with different values of whatever property is being “observed”.)

This relative state changes whenever the observer learns something about the system – and since since this results in the probabilities of all values other than the one observed becoming zero (while the one experienced by the observer becomes certain from the observer’s point of view) it is sometimes called “collapse” – though it should be noted that the total of all the probabilities remains the same, so it might be better to think of the probability distribution being collapsed “sideways” onto the observed value rather than “down” to zero everywhere.

As an example, consider the case of an electron that has just passed through a selector that ensures that its spin in the z-direction is positive (let’s call this “up”). If we think of the spin of the electron as a system in its own right and if we can control the path of the electron without interacting with its spin, then we can represent (what we know about) the spin by a quantum state, and it turns out that if we know the spin is up then we cannot assign any particular value (say left or right) to any horizontal component. All we can say is that if we pass many such electrons through another vertical spin selector they will all measure spin up, but if we measure in any horizontal direction we’ll get positive and negative results with probability 1/2 each. BUT (and here’s the “collapse”) if, on doing that horizontal measurement, we observe spin (say) left, then when we subsequently measure the same horizontal spin we’ll get the same result with certainty and the probability of seeing right will have gone down to zero (while if we go back to measuring the vertical component it will now have equal chances of being up or down – which is another reason for not liking the “collapse” language since what collapses one distribution has spread out another, and in particular has raised the chance of seeing down next time from 0 to 1/2).