Month: January 2022

Einstein’s train embankment thought experiment (showing relativity of simultaneity)

The fundamental observed fact that underlies this is that all unaccelerated observers get the same result when measuring the vacuum speed of light in any direction.

So if an observer, A, riding at the middle of a long train sees signals emitted from the front and back of the train at the same time, since the distances and speeds are equal he or she infers that they were emitted at the same time.

If A raises a flag if and only if he or she receives the signals at the same time, then an observer B on the bank also knows that the signals reached A at the same time. But when raising the flag, A has moved ahead in B’s frame from where he or she was when the signals were sent. So B sees that the signal from the back travelled further than that from the front and so must have taken longer (since B sees exactly the same speed of light in all directions as A). So from B’s point of view, the signals which A saw as being emitted simultaneously were actually not simultaneous (with the one from the back having been sent earlier).

Source: (152) Alan Cooper’s answer to Does the train embankment thought experiment of Einstein really demonstrate relativity of simultaneity? Is there actual experimental proof of RoS? See comment attached to the question for details. – Quora

How can time be dilated for both observers without contradiction? – Quora

Because it’s not the timing of the clock of either observer that is dilated but the other observer’s interpretation of their view of it. It’s really no more of a contradiction than the fact that if we are facing one another then when I ask you to raise your right hand you raise the one on my left. (Note: I didn’t say it’s the same, just “no more of a contradiction”. If it seems mysterious to us that’s just because our experience of comparing notes at low relative velocities leads us to wrongly guess that there is an absolute standard of simultaneity for spatially separated events.)

Note: It’s only when both observers are in fixed inertial frames that they both see the other’s time as dilated. In the travelling twin scenario (appended as a comment to the actual question) one or other of the twins must undergo acceleration (or impulse) in order for them to get back together, and whoever is accelerated (or jumps from one frame to another) sees much of the time of the other as very much compressed (or collapsed to a single point of his or her own time in the physically impossible case of an instantaneous turn-around).

This is a real asymmetry because in the absence of gravity (ie in Special Relativity), the twin who is accelerated actually feels a force which the other does not. So they do know which one was accelerated.

In General Relativity an observer in free fall can accelerate without feeling any forces (and if the traveler is turned around by a gravitational slingshot then the time difference is explained by the time spent at the bottom of a very deep gravitational potential well – or equivalently the proximity of a very large mass – which is again an asymmetry between their experiences); but that’s another matter form the question of whether it’s a “paradox” in SR. In SR one cannot accelerate without being pushed by something like the reaction forces of a rocket and one twin knows he or she is in a rocket ship and feels the force while the other does not.

Source: (129) Alan Cooper’s answer to How can time be dilated for both observers without contradiction? – Quora

What is the problem associated with solving Klein-Gordon partial differential equations? – Quora

There’s no problem with solving the Klein-Gordon Equation. It’s just that some of the solutions have a form for which [math]H\Psi=i\frac{d\Psi}{dt}=E\Psi[/math] with arbitrarily large negative values of [math]E[/math]. If we were to try to interpret the solution as a quantum wavefunction this would correspond to having no lower bound on the energy – which is physically unstable.

Source: (128) Alan Cooper’s answer to What is the problem associated with solving Klein-Gordon partial differential equations? – Quora

Entanglement

Entanglement is just a word we can use to describe a situation where knowledge of some property of one object gives us information about some (possibly different) property of another.

The term is rarely used in the classical case, because we take it for granted. If we separate a pair of gloves for example and pack them up in identical boxes and then choose by a coin toss to send each in one of two opposite directions, then we are not surprised by the fact that if someone who knows how they started out but does not know the result of the coin toss opens one box and sees a left glove, he or she knows immediately that whoever opens the other box will see a right glove.

There is often similar classical chance-based uncertainty in our knowledge of quantum systems; but for such systems, even in the most precisely prepared “pure” states the knowledge of some properties makes it impossible for us to know others. This residual uncertainty is expressed by representing the state of the system by a “state vector” in a Hilbert space and the “mystery” of quantum entanglement is that the correlations between systems (like the gloves) that were once together but are now far apart can sometimes be greater than would be possible for any way of randomly assigning the properties at the outset.

The reason this extra correlation is sometimes considered “spooky action at a distance” is because the change of state vector (often called “collapse”) that occurs when we open one box seems to trigger a simultaneous collapse at the other box – and in a way that can change what the remote observer will see when looking at different properties from the one that obviously has to be opposite. At first sight it may seem that this effect might be used to send a signal where what the second observer sees might depend on what the first one chose to measure, but that turns out not to be the case.

Whether or not this bothers you may depend on whether you consider the state vector to be a property of the system itself or rather of the way it appears to a particular class of observers.

Is special relativistic time dilation a real effect or just an illusion? Given two inertial frames each observer finds that the clock of the other runs slower than that observer’s own clock. So who is right? 

This is a pretty good answer except that I wouldn’t say either of them is right if they think that their perception of relative slowness represents something that is objectively true for all observers.

Time dilation is a real effect on the perceptions of observers (with regard to the rates at which one another’s clocks are ticking). Neither of them is “right” if they think there is any real sense in which the other’s clock is objectively slower. But neither of them is wrong about how it appears to them, so it’s not really an illusion any more than the fact that if they are looking at one another then their ideas of the “forward” direction are opposite to one another. What turns out to be more of an illusion is the sense we all have that there is some absolute standard of time which determines which of two spatially separated events occurs before the other.

(3) Alan Cooper’s answer to How can a moving/faster observer see a ‘stationary’ clock running slower, when it doesn’t, e/specially when his really does? All the YouTube videos about the Twin Paradox say so. In what sort of universe is that possible? – Quora

The relativity principle does NOT say that all frames are equivalent, just the inertial (unaccelerated) ones. In the twin “paradox” situation, the traveler and stay-at-home both figure that the other is ageing more slowly while their relative velocity is constant but when the traveler turns around he or she figures that the stay-at-home suddenly ages much more rapidly. The two may have different explanations in mind for the effect, but when they get back together they agree on the result.

Source: (3) Alan Cooper’s answer to How can a moving/faster observer see a ‘stationary’ clock running slower, when it doesn’t, e/specially when his really does? All the YouTube videos about the Twin Paradox say so. In what sort of universe is that possible? – Quora

(3) Alan Cooper’s answer to How does the twin (clock) paradox (in SR) really work? Please see the comment for the specific case. – Quora

According to the comment, what this question is really asking us to address is something completely different from “How does the twin (clock) paradox (in SR) really work?”

What the comment asks us to explain is as follows: “A clock flies around the equator eastwards, it ages slightly less. A clock flies around the equator westwards, it ages slightly more. Than a clock which stayed, at home.”

The explanation for this (in the context of either Special OR General Relativity) is that the alleged effect exists only for motion relative to the Earth’s surface (which is already rotating). So if I stand still on the Earth’s surface I will age more slowly than a twin who stands still relative to the Earth’s centre (which entails flying Westwards at a rate of about 1000mph) and more quickly than a twin who flies Eastwards and so is actually moving more quickly relative to the Earth’s centre.

Note: As with the usual out-and-back twin “paradox” what makes the travelers different from the stay-at-homes is that they are not in fixed inertial frames but are accelerated. And in the case of this scenario the twin flying West at 1000mph is basically staying stationary wrt the Earth’s axis which (if we neglect the curvature of the Earth’s orbit) is pretty close to being in an inertial frame. But the one standing still is actually rotating with the Earth and so is accelerating (centripetally) towards the axis. And the one flying East is actually accelerating even more. According to SR, an observer who is accelerated figures that stationary clocks towards which she is accelerating are speeded up by an amount that more than counterbalances the fact that if the speed was fixed she would figure that they were slowed down. And if you do all the calculations it turns out that when they get back together they all agree on their relative ages (with Eastflier younger than Standstill and Standstill younger than poor old Westflier(who is actually the only one who is really standing still)).

Source: (3) Alan Cooper’s answer to How does the twin (clock) paradox (in SR) really work? Please see the comment for the specific case. – Quora

(3) Alan Cooper’s answer to What is the Calabi-Yau manifold? Are we ‘inside’ it right now or is it just a mathematical concept? – Quora

Calabi-Yau manifolds (there’s not just one) are a type of mathematical concept, but they’re not “just” that as they do have applications in certain attempts to describe physics. The role they usually play in physics is to help us formalize the relationships that we postulate between the various internal variables that describe what particles are likely to show up at a point in space time. As such the theory often combines the six dimensions of a Calabi-Yau manifold with the four dimensions of the space-time that we are ‘inside’ to get a total of ten dimensions. But the extra dimensions are often either considered to be very small in some sense, or to have the part that contributes to the physics we see be just a slice through the whole thing. In the first case it makes more sense to say (as in James Bridgeman’s answer) that there’s a C-Y manifold at every point inside us (rather than vice versa), and in the second case that the entire space-time we live in is just a (4d) slice through the extended C-Y manifold (with other slices or “branes” perhaps corresponding to “alternate universes” of some kind). But neither of these cases is in any sense known to be true. So far it’s all just speculative construction of mathematical models that might eventually be shown to describe our actual physics.

Source: (3) Alan Cooper’s answer to What is the Calabi-Yau manifold? Are we ‘inside’ it right now or is it just a mathematical concept? – Quora

(3) Alan Cooper’s answer to Why is the Raman effect’s classical interpretation not adequate? – Quora

The classical theory of the Raman effect is not adequate for determining the actual spectrum because it allows the molecule to have arbitrary amounts of vibrational (or rotational) energy and in fact the possible energy levels are quantized (just like everything else in physics).

Source: (3) Alan Cooper’s answer to Why is the Raman effect’s classical interpretation not adequate? – Quora

(3) Alan Cooper’s answer to Does any physicist truly understand wave function collapse? – Quora

There are certainly many physicists who use the term “wave function collapse” to refer to something that they understand well enough for their own purposes. Whether or not that means they “truly understand” it depends on what you mean by that (in my opinion rather silly) expression.

Source: (3) Alan Cooper’s answer to Does any physicist truly understand wave function collapse? – Quora