## What does it mean to say that there is a distance between two events in time? (Another Quora Question)

It probably means that the speaker is taking a Galilean approach to physics.
In modern relativistic physics, the property of having a time-like separation between two events is independent of observer, but the magnitude of that separation depends on the observer. And two events which are spacelike separated, while having no time difference for some observers, will still appear to have a non-zero time difference for others.
So the concept of a time (or space) “distance” (ie a specific value of the difference) between two events in space-time does not make sense without reference to an observer.
However in the case of two time-like separated events the time difference is nonzero for all observers, and if we restrict to inertial observers it has a nonzero minimum (which corresponds to the time difference as seen by an observer who experiences both events directly without any intervening acceleration or gravitational field gradient). But although this minimum is in principle computable by any observer it does not correspond to the time difference actually “seen” by that observer.

## Are Bayesian and Frequentist probability really inherently completely different, or is there a false dichotomy between them?

I would say that they are inherently “somewhat” different even though they are both used for similar purposes and obey the same mathematical axioms (of a countably additive non-negative measure on some space of “outcomes” in which the measure of the set of all outcomes is 1).

Of course there are many probability measures in the mathematical sense which have nothing to do with any actual probabilities in the real world, so having the same mathematical structure does NOT preclude their being “completely different”.

But probabilities in the real world are supposed to have something to do with our expectations regarding what will happen when we perform an experiment which could in principle be repeated an unlimited number of times. And since any concept of probability is intended as a measure of how “likely” we expect physical events to be, if both kinds are useful they can’t really be all that different in what they lead us to expect.

The difference, as I see it, is as follows:

The frequentist probability of an “Event” (which is just the jargon for a subset of the space of all possible outcomes) is assumed to be the proportion of “all” experimental trials in which the outcome is consistent with the event. Despite its rather fuzzy definition (in terms of whatever we mean by “all” trials), this is something that is assumed to be an actual property of the experiment, albeit one that can never be determined by experimental tests (because any finite set of trials might fail to be representative). Frequentist statisticians often try to choose between different possible sets of assumptions by using each set of assumptions for computing the probability of what they have actually seen and choosing the assumptions that lead to higher probability, but they generally do so with the mindset that there is one set of assumptions that is really true.

Bayesian probability, on the other hand, is something that is always acknowledged to depend on the observer (through both initial choices and subsequent acquired knowledge). Given those choices and experience, the Bayesian probabilities are perfectly well known because the Bayesian approach provides an explicit rule for modifying one’s assumptions in the light of experiments. But because it depends on the the observer’s assumptions and experience, the Bayesian probability is not a well-defined property of the experiment and outcome alone.

The difference is subtle though, because one cannot do any frequentist analysis without making an assumption (usually about what constitutes a set of “equally likely” outcomes). But though one can make and change the assumptions, the Frequentist attitude remains that they are assumptions about something that exists and are either true or false, while the Bayesian does not necessarily ever claim that there is one set that is really “true”.

However there are theorems which tell us (roughly speaking) that if some frequentist assumption is in fact correct, then no matter how bad the initial assumptions of the Bayesian are, after a lot of experiments it is very probable (in the frequentist sense) that the Bayesian probabilities become close to the “correct” ones.

So in the end both approaches end up giving the same advice about what one “should” expect (though neither gives anything I can understand as a non-circular definition of what that means!) and whether the difference in attitude is a “false dichotomy” is something I think we each have to decide for ourselves.

## Given that the Lorentz transformation is symmetrical with respect to interchange of space and time, how does it lead to length contraction but time dilation?

This is a question that I am surprised to not have seen before (especially since I have had to remind myself of the answer more than once – including, I suspect but can’t be sure, from way back before I entered my dotage).

It is true that in one space dimension the transformation equations

$x’=\gamma(x-\beta t)$ and $t’=\gamma(t-\beta x)$

are completely symmetrical with respect to interchange of $x$ with $t$ and $x’$ with $t’$.

(and in the case of three space dimensions the same applies if $x$ and $x’$ are the coordinates in the same direction as the relative velocity, so it’s not got anything to do with the dimension).

So what is the difference?

Well here it is in a nutshell.

When we measure the length of a moving measuring rod, we look at both ends at the same time and so are looking at the spatial distance between two events at the same time in our frame of reference.

But when we measure the time between two ticks of a moving clock we are looking at the time difference between two events that are NOT at the same spatial position in our frame.

So the nature of the two measurements is not symmetrical with respect to interchange of space and time.

I may add some more explanation and diagrams to show how this does lead to contraction for the rod length and dilation for the tick interval, but I wanted to get this off my chest right away – and also to address a couple of natural follow-up questions.

Namely, what kind of measurements would give the symmetrical outcome? Are there situations in which these others might be relevant? And why do we instinctively prefer the ones we do?

So, for example, what kind of time measurement would be symmetrical compared to our usual rod length measurement (and so would give a “time contraction” rather than the usual time dilation)?

Since the rod length involves looking at both ends at the same time in our frame, the corresponding time measurement would involve looking at the interval between two ticks at the same place. But how can we do this if the clock is moving? Well we could if the clock was extended in space, and if we have a long train of clocks that are synchronized in their own frame, then you can easily check that observers who look at the time between the ticks right in front of them will actually see a shorter interval than that measured by the travelling system – ie a time contraction.

And going the other way, what kind of measurement would give a length dilation? Well that would have to be the symmetric version of our usual clock measurement. And corresponding to our usual measurement of the time interval between two ticks at the same place in the moving clock’s frame, interchanging space and time would have us measuring the spatial distance between events where the two ends of the rod are at the same time in the rod’s frame. For example the managers of the rod might set off flares at both ends in a way that they, travelling with the rod, perceive as simultaneous. If we measure the distance between where we see those two flares then it will indeed appear dilated relative to the length of the rod in its own frame.

So now we come to the final question. Is there anything really “wrong” about these alternative kinds of measurement? If so what is it? Or is there just something about us which makes us think of what we do as natural and the alternative as somehow, if not actually wrong, then at least rather odd?

Here’s what I think (at least for now). The thing that makes us prefer to measure lengths in terms of events at the same time in our frame but times in terms of events at the same place in the moving frame is the fact that we, as blobs of space time, are much more extended in time than in space. (This is evident in the fact that we live for many years but do not extend for many light years in our spatial extent – or equivalently that in units adapted to our own spatial and temporal extent the numerical value of c is very large.)

So here’s a follow-up question. Could we imagine an entity which was the other way around? (ie of brief duration but of great spatial extent) And from the point of view of such an entity would it make sense to define measurements differently (as suggested above to achieve the effect of time contraction and length dilation)?

OR is it more just a matter of causality?

P.S. This is a question and answer that I have been meaning to post for some time, but was prompted to do so by Domino Valdano’s excellent answer to another question (in which she covers pretty much the same ground with a slightly different way of expressing the ultimate reason for why we measure as we do – which I may yet end up deciding that I prefer to my own). Please do read that one too!

## Yet Another Quora Question:

### Is there experimental evidence of RoS (relativity of simultaneity) available? According to Carl Popper something that cannot be falsified is metaphysics. Can the ALT (absolute Lorentz transformation) and absolute simultaneity be used instead of RoS?

Yes. There is lots of evidence that IF observers define simultaneity of remote events by comparing the arrival times of light signals from those events and adjusting for the light travel times, then they will NOT agree on which pairs of events are or are not simultaneous.

What cannot be falsified (and so, according to Popper, is not worthy of consideration as scientific) is the claim that one particular set of “stationary” observers are “correct” and the rest are “wrong” about the “actual” simultaneity (with the observations of the “wrong” ones being explained by effects of their “motion” on their clocks and measuring rods).

The “metaphysical” question of whether or not there is such a particular set of “stationary” frames is generally resolved by reference to “Ockham’s Razor” which advises us to prefer an explanation which involves fewer arbitrary choices. To the extent that the identification of a particular preferred set of “stationary” frames is arbitrary, we are therefore advised to treat them all as equally valid and make no choice of what constitutes “absolute” simultaneity.

Of course, any particular feature of the universe (such as for example, the cosmic background microwave radiation (CBMR)) can, if we like, be taken as defining an “absolute” rest frame (and so an “absolute” sense of simultaneity). And if I were moving at a noticeably high speed relative to the CBMR, then it might indeed be presumptuous to declare my own synchronization as more fundamental, but it would not in fact lead to any difference in anyone’s prediction of any testable event.

## More on acceleration in the Twin “Paradox”

There is a possibly interesting comment thread following my answer to How exactly do Minkowski diagrams prove that acceleration is not needed in resolving the Twin Paradox in Special relativity? – Quora

In it Peter Webb passionately (and sadly sometimes rudely) defends the position that “acceleration has nothing directly to do with the Twin paradox”. This is a position shared by a not insignificant minority of apparently competent physicists (on Quora, Brent Meeker comes to mind as a prominent example), but although I continue to feel that the issue is vastly overblown, I find that the “it’s not acceleration” view is misguided and at least some of its proponents are sufficiently intransigent and aggressive that they demand a further rebuttal.

So, with reference to Peter’s final summary of his position, here goes:

The “paradox”/interest in the TP is that it demonstrates time dilation. The different ages merely demonstrates this.

To be frank, I have no idea what he is talking about here. The usual time dilation of special relativity, which applies in the case of two inertial observers having a constant relative velocity with respect to one another, only exists in the understanding of each observer regarding the clock of the other. There are many ways of demonstrating this effect (most famously with muons from cosmic ray interactions with the upper atmosphere, but also in many other ways), but it is not something that actually happens to either observer in any objective sense – at least not while they both continue in an inertial state of motion.(We may think its obvious that the muon is the one with a dilated lifetime but from the muon’s frame of reference the Earth looks like a very flat pancake and the distance it has to travel can be covered well within its lifetime.)

The Twin “Paradox”, however, is something different in that it does lead to an objective fact of the matter regarding which twin or clock aged more between two specific events in space-time.

Maybe 80% of people with some interest in science think that time dilation is a consequence of acceleration. It isn’t, directly. It is a result of changing reference frames and in the Twin’s paradox that involves acceleration. But time dilation still occurs in the absence of acceleration (eg 3 brothers), and it is easy to show that identical acceleration profiles produce very different amounts of time dilation.

I am not aware of the statistics regarding what “people with some interest in science” think, but I doubt that any significant percentage of them think that the symmetric relative time dilation of unaccelerated motion is a “consequence of acceleration”.

But on the other hand, if they think that the objective observable time difference in the Twins “Paradox” is a consequence of acceleration, then I’m right there with them! (I have no position on how “directly” consequential the acceleration is, but I am happy to see an acknowledgement that indeed having one twin change frames does involve acceleration.)

But now we come to the 3 brothers.

It is true that they provide yet another way of confirming the relative time dilation effect. Indeed, the incoming “brother” brings back to Earth a record of what the outbound one’s clock said when they met  (though of course this is not necessary, and the information could just as easily have been passed by a radio signal). But he (or the radio message) could also inform Earth of the time that the outgoing traveller inferred was on the Earth clock at his idea of when the crossing of paths happened. This would be consistent with the usual unaccelerated time dilation, which is perfectly symmetrical in the sense that while the Earth observer thinks that at the event on the traveller’s path which is concurrent with any time $t_{E}$ on his Earth clock (measured from the outbound traveler’s departure event), the outbound clock reads $t_{O}=\frac{t_{E}}{\gamma}$, and the outbound observer thinks that at the event on Earth which is concurrent with any time $t_{O}$ on his travelling clock (measured again from the shared departure event), the Earth clock reads $t_{E}=\frac{t_{O}}{\gamma}$.

So, at the particular event where the travelers meet, if the Earth brother thinks this occurs at time $t_{M,E}$ then the outbound traveler’s clock reads $t_{M,O}=\frac{t_{M,E}}{\gamma}$ and he thinks that the concurrent time on the Earth clock is just $\frac{t_{M,O}}{\gamma}=\frac{t_{M,E}}{\gamma^{2}}$.

The inference of the inbound traveler on the other hand, after synchronizing his clock with the outbound, is that during the rest of his trip to Earth, the Earth clock only advanced by the same $\frac{t_{M,E}}{\gamma^{2}}$.

But when they synchronize clocks the two travellers can also compare notes on what they think is showing on the Earth clock at that time. And their disagreement on that will show them immediately that everything works out and there is no paradox.

So in the 3 brothers case there is clearly no paradox, as everyone is aware that each traveler considers only a small part of the Earth’s experience as concurrent with their own travel time and there is no reason to expect that those two intervals should add up to the whole time on Earth. So no two observers are ever forced to agree that only one of them is right about the time dilation of the other.

Of course there is not a paradox in the single traveler case either, but it is a more powerful (wrong) intuition that the two intervals judged by the traveller to be concurrent with the legs of his trip should cover the entire Earth time interval (even though we now know from the 3 brothers analysis that they do not).

Note: I am not disputing the validity of the 3brothers scenario as the most appropriate explanation of why there is no paradox. But I will continue to insist that it is not significantly counterintuitive in its own right and that the solution it provides to the single traveler version identifies the “missing” Earth time with the traveller’s “frame jump” and that his “frame jump” (no, not just in his head but between two different frames that he is actually at rest with respect to) is nothing more than an integrated acceleration.

This leads to the question of what we can learn from the sudden jump in the traveler’s inferred Earth time at the turnaround. We learn that when a traveller changes frames  then, as his simultaneity space changes, the times he considers concurrent on clocks that are remote from him in his direction of velocity change are advanced (and those behind retarded).

Of course the sudden turnaround is physically impossible and in any real situation there would be a period of finite acceleration. But it is a simple exercise in calculus to approximate a period of finite acceleration with a number of discrete jumps and conclude that while the traveler is turning around his inferred current time on Earth advances more rapidly than when he is not accelerating (at a rate proportional to both his acceleration and his distance from Earth).

Now none of this made any use of General Relativity. The acceleration effect we have discovered is purely from Special Relativity and I do agree that it’s a monstrous abuse to address the Twin “Paradox” by invoking acceleration=>GR=>”it’s like being in a gravity well which we all know (from the movies!) causes time dilation”.

But the real beauty of all this is that it DOES go the other way!

In SR, acceleration=>time dilation (relative to clocks that are remote in the direction of acceleration), and so, using only the equivalence principle, we learn (without any of the harder GR analysis) that Matt Damon really does outlive his grandchild!

Given Peter’s earlier answer about the Ehrenfest “Paradox” I am totally surprised and disappointed that he doesn’t seem to appreciate this.

## What is Schrodinger’s equation? Is it deterministic or not? If it is, how can we prove that? And what conditions must be satisfied for it to be non-deterministic?

Schrodinger’s equation was originally just the partial differential equation satisfied by the position-space wave function of a particle (or more general system) in non-relativistic quantum mechanics. The same name is sometimes also used for the equation $\frac{d}{dt}\Psi(t)=iH\Psi(t)$ satisfied by the state vector in any NRQM system regardless whether or not a position-space representation is being used (or is even available).

It is deterministic (in the sense of determining $\Psi(t)$ uniquely for all $t$ if given an initial condition $\Psi(0))$, so long as the Hamiltonian $H$ is self-adjoint (symmetry is NOT enough!).

The proof of this involves more analysis than I could fit into a Quora answer, but in the general case it follows from the fact that for any self-adjoint operator $H$ on a Hilbert Space, the equation $\frac{d}{dt}\Psi(t)=iH\Psi(t)$ is uniquely satisfied by $\Psi(t)=e^{iHt}\Psi(0)$ where the complex exponential of $H$ is defined in terms of its spectral resolution; and for the PDE special cases it might be done by various theorems involving greens functions or Fourier analysis and convergence properties of improper integrals.

It may be non-deterministic if $H$ has not been specified on a large enough domain to be essentially self-adjoint (as sometimes happens if boundary conditions are omitted from the specification of a problem in which the particle is confined somehow – either by an infinite potential or by living in a single cell of a crystal lattice for example). But such cases are normally just due to inadequate specification of the problem rather than to a real physical indeterminacy.

So I would say that in a properly defined quantum theory model the Schrodinger equation is indeed almost always deterministic.

[N.B. It wasn’t part of the actual question, but I should perhaps add that the reason this does not make quantum mechanics deterministic is because even complete knowledge of the quantum state of a system is not sufficient to predict the outcomes of all possible experimental measurements. For any a state which happens to produce a predictable value for one observable there will be other observables for which the outcome is uncertain.]

## What is the definition of an eigenstate of a hermitian operator?

An eigenstate of a quantum observable is a state which results from a measurement of that observable which has produced a precise value; and according to quantum theory this means that it is represented by a normalized eigenvector for the corresponding self-adjoint operator (whose eigenvalue is equal to the observed measurement value).

An eigenvector of an operator $A$ is a vector $\Psi$ for which $A\Psi=\lambda\Psi$ for some number $\lambda$ (which is called the corresponding eigenvalue).

## Falling from Space

If it runs out of fuel then its free fall is just the current orbit in which it will stay essentially forever (since it is well above any atmospheric drag that might slow it down). But in the unlikely event that it has the extremely large amount of fuel needed to kill all of its orbital velocity, (and if it actually uses the fuel for that purpose), then it will fall straight down to the earth in a time that one can calculate – either with a bit of calculus, or by the clever trick of applying Kepler’s law for a very thin ellipse of major axis length equal to the orbital distance.

[Kepler’s third law says “The square of a planet’s orbital period is proportional to the cube of the length of the semi-major axis of its orbit” so the period is proportional to the 3/2 power of the major axis; and the full axis of a narrow ellipse from the satellite down to the earth is just the radius (ie half the major axis) of its previous circular orbit. So the period of that ellipse, if completed, would be the geostationary period of 24 hours divided by a factor of 2^{3/2}. But the satellite only gets (a bit less than) halfway round before crashing, so the actual time for the fall is a bit less than 24/2^{5/2}=3\sqrt{2} or about 4 hours.]

P.S. For the Moon, since the orbital period is 28 days, the time to fall would be more like 5 days, but I think the claim attributed to Galileo may arise from a misreading of his argument that the Earth is spinning on its axis (basically something like: If the Earth was NOT rotating and instead everything on the celestial sphere was going round once every 24 hours, then the Moon, Sun, and stars might all be at the geostationary distance but only those near the equator would actually be in balance and those at the pole would just fall to Earth in 4 hours)

P.P.S. It is not clear that Galileo actually had a good understanding of orbital mechanics and the inverse square law, so if he estimated a falling time of 4 hours that may have been little more than a lucky guess

## fromQuora: Isn’t a ‘probability wave’ simply a statistical function and not a real wave? Does it no more ‘collapse’ than me turning over a card and saying that the probability ‘wave’ of a particular deal has collapsed?

Well, as other answers have noted, the wave function of quantum mechanics is not a probability wave as its values are complex and it is only the squared amplitude that gives a probability density. But the process of “collapse” involved in a quantum measurement does involve something like your playing card analogy.

There are actually two stages in the measurement and observation process. One is the interaction with an incompletely known measurement apparatus which reduces or eliminates the prospects for future interference and basically turns the previously pure state of the isolated system (considered as a subsystem of the larger world) into a statistical mixture. And the second is the collapse of that statistical mixture by observation – with the result that, from the observer’s point of view, of the many possible alternatives only one is actually true.

And if this makes it seem to you that the “state” of the system actually depends on the observer then you are on the right track. (But it is nothing special about the “consciousness” of the observer that is relevant here. Almost any localized system could play the same role relative to the rest of the universe.)

Any configuration history of any physical system can be considered as “seeing” the rest of the universe in a “relative state” which “collapses” when the configuration history in question passes a point beyond which the configuration includes information about that particular measurement value.