special relativity - What does the accelerated twin see in the twin paradox?

I am referring to the usual twin "paradox" where one twin remains on Earth while the other one takes a round trip journey on a rocket. I understand how the experience of the twins is not symmetrical, because one is accelerating, and that the solution involves changing reference frames. What I haven't seen in common answers, and what I am curious about, is what would one twin (on the rocket) see if she could constantly look at the other twin's clock. Imagine that the travelling twin had a super telescope and watched the Earth-bound twin's clock constantly during her trip. This scenario includes the following stages:

1. Positive acceleration, +a, away from Earth


2. Floating freely for a while at velocity +v


3. Negative acceleration, -a, to slow to a stop relative to Earth


4. Briefly motionless relative to Earth (v=0)


5. Accelerating towards Earth at -a until reaching velocity -v


6. Floating freely towards earth at velocity -v



7. Final positive acceleration, +a, to land on Earth (v=0).

What would the twin on the rocket see on Earth, looking from afar:

1. While accelerating away?


2. While floating freely before starting the deceleration?


3. While decelerating but still moving away?


4. During transition (turn around)?


5. While decelerating but now moving back towards Earth?


6. While floating freely back towards Earth?



7. While accelerating to a stop on Earth?

Imagine the traveling twin constantly looking through the telescope and comparing the Earth clock to her rocket clock. I'm looking for an answer like "...now the Earth-clock appears to be ticking slower than her own clock on the rocket". And "...at this point the clocks are momentarily showing the same time." You can keep the math to the bare minimum, in fact no math is best! Thank you in advance.

Answer

I made some videos about this a few years back, on YouTube. It shows what the traveler sees on a variety of clocks in the scene, for the entire duration of the journey.

I would suggest "Earth gravity front view" to start with ("Earth gravity" represents the value of acceleration in this context!)

The terse explanatory text is reproduced here:

The twin "paradox" (in quotes because it is NOT a real paradox!) is a basic teaching scenario for Special Relativity: http://math.ucr.edu/home/baez/physics...

This is a series of visualizations of the journey from the point of view of the traveling twin, who flies 20 light years away from his home station then returns. The journey consists of four parts, joined together. The first quarter is an acceleration away from the station. During the second and third quarters the ship accelerates towards the station (so that at the half way point the ship is stationary 20 light years away). In the fourth quarter the ship accelerates away from the station in order to come to rest there.

The total coordinate travel time (shown as a red dot in the top left HUD clock) is 43.711/58.918 years for acceleration at earth/moon gravity levels, whilst proper time (green dot) is 12.101/38.694 years. The yellow dot represents the time the traveler would see on the station clock face through a very powerful telescope! The octahedral stations (spaced one light year apart and one light year to the left of the flight path) are all synchronized to coordinate time and rotate once over the course of the whole journey. There is a 2x2 light year wall one light year beyond the far end of the journey, and large rectangular frames every 5 light years. Where a floor is shown it is 1 light year per stripe, and there are small 1 ly milestone spheres along the way, with a larger one every five light years.

The flights are rendered without relativistic effects and then for two values of acceleration (currently earth gravity and moon gravity). The distortion artifacts are due to aberration of light, the Doppler effect and the headlight effect. The earth gravity videos exhibit some nice penrose-terrel "rotation" effects. Magenta markers in a circle show where you really are in the scene (in the sense that things outside the circle are behind you, whilst those inside the circle are in front of you. Doppler shift = gamma for this circle), and grey markers show the circle where the doppler shift is 1 (gamma is in theory directly observable on this circle).

electrostatics - Why does a dielectric slab move inside the capacitor?

The electric field is in Y direction.Therefore shouldn't force be also in the vertical direction.Also if electric field is y direction then what's the force that is pulling the slab inside the capacitor?

I am not asking to calculate the force.I am just asking what's the origin of this force

quantum field theory - Resources for 2 Particle Irreducible (2PI) or Cornwall-Jackiw-Tomboulis (CJT) formalism

I'am currently learning the 2 particle irreducible (2PI) or Cornwall-Jackiw-Tomboulis (CJT) formalism. Does anybody know a textbook or a review that treats this subject? As far I only found the original papers like http://dx.doi.org/10.1103/PhysRevD.10.2428 but no modern review. Of course there are a lot of papers that use this formalism, but I haven't found a general introduction yet.

As far I know how to compute the generating functional, but not why this is a useful approximation, where it can be used (limits of the framework) and so on. At the end I am interested in renormalization in the Hartree approximation (I have resources for that, but not for the "beginning")

Answer

There is an extensive, survey-like introduction by Juergen Berges, dating from 2004. For the renormalization of 2PI effective actions see also a very recent paper by Carrington et al..

fluid dynamics - Could Navier-Stokes equation be derived directly from Boltzmann equation?

I know how to derive Navier-Stokes equations from Boltzmann equation in case where bulk and viscosity coefficients are set to zero. I need only multiply it on momentum and to integrate it over velocities.

But when I've tried to derive NS equations with viscosity and bulk coefficients, I've failed. Most textbooks contains following words: "for taking into the account interchange of particles between fluid layers we need to modify momentum flux density tensor". So they state that NS equations with viscosity cannot be derived from Boltzmann equation, can they?

The target equation is $$ \partial_{t}\left( \frac{\rho v^{2}}{2} + \rho \epsilon \right) = -\partial_{x_{i}}\left(\rho v_{i}\left(\frac{v^{2}}{2} + w\right) - \sigma_{ij}v_{j} - \kappa \partial_{x_{i}}T \right), $$ where $$ \sigma_{ij} = \eta \left( \partial_{x_{[i}}v_{j]} - \frac{2}{3}\delta_{ij}\partial_{x_{i}}v_{i}\right) + \varepsilon \delta_{ij}\partial_{x_{i}}v_{i}, $$ $w = \mu - Ts$ corresponds to heat function, $\epsilon$ refers to internal energy.

Edit. It seems that I've got this equation. After multiplying Boltzmann equation on $\frac{m(\mathbf v - \mathbf u)^{2}}{2}$ and integrating it over $v$ I've got transport equation which contains objects $$ \Pi_{ij} = \rho\langle (v - u)_{i}(v - u)_{j} \rangle, \quad q_{i} = \rho \langle (\mathbf v - \mathbf u)^{2}(v - u)_{i}\rangle $$ To calculate it I need to know an expression for distribution function. For simplicity I've used tau approximation; in the end I've got expression $f = f_{0} + g$. An expressions for $\Pi_{ij}, q_{i}$ then are represented by $$ \Pi_{ij} = \delta_{ij}P - \mu \left(\partial_{[i}u_{j]} - \frac{2}{3}\delta_{ij}\partial_{i}u_{i}\right) - \epsilon \delta_{ij}\partial_{i}u_{i}, $$ $$ q_{i} = -\kappa \partial_{i} T, $$ so I've got the wanted result.

quantum mechanics - Why should the Planck constant be a constant throughout all space?

Our value for the Planck constant $h$ can be found on experiments on Earth, but how do we know that the Planck constant doesn't change throughout space, for instance it depends weakly upon the curvature of spacetime? As far as I know we have only done experiments to determine the value of $h$ close to the Earth's surface, so I don't think there's any direct evidence to suggest it should be a constant throughout all space, so I proposed the idea to my friends.

One called me a nut, and said that the value for Planck constant has to be the same throughout all space otherwise the conservation of energy would be violated. He put forward the following situation: Suppose that in one region of space the Planck constant is found to be $1$ in SI units (call this Space $A$), and you have a photon of frequency $1$ Hertz. This photon then traveled to a different region in space where the value for the Planck constant is now $3$ in SI units(call this Space $B$). Hence the photon has gone from having $1$ Joule of energy to $3$ Joules of energy (by the relation $E = hf$). In his own words, a twitch of a finger here can cause an explosion in a different region in space. To this I replied: So what? As far as I know the conservation of energy argument derives from the fact that space looks the same everywhere we look, and if the Planck constant was really different in different areas of space, then that argument shouldn't hold, and conservation of energy shouldn't hold either. Even if I did want to abuse this system by creating a feedback loop where the energy got larger and larger, the point was if I ever tried to extract energy from Space $B$ back to Space $A$ it would become a small amount of energy again since the value for $h$ decreases.

Another friend called me a nut, because he believed that in sending the photon from Space $A$ to Space $B$, I would violate the Second Law of Thermodynamics by decreasing the overall entropy. He didn't really expand on it, and (I believe) neither of us have the abilities to calculate the actual entropy change, so this point remains unresolved.

Finally, another friend called me a nut, because he believed that if the value for $h$ was different in different regions of space, then all our calculations ever done in physics dealing with the stars and their brightness and so on and so forth were all incorrect, and that would be troubling. To this I replied that firstly, the changes in the value for $h$ could possibly be very small for 'normal' regions of space, and secondly that there was nothing inherently WRONG about all these other calculations we've done being wrong, in fact we know they are wrong already because we can't account for either dark matter or dark energy! So in my mind this wasn't a really good argument as to why the Planck constant couldn't be variable over space.

With all that being said, I'm still fairly certain that the Planck constant is a constant over space, because it is called a constant. So is there an error in my previous arguments, or what is the conclusive reason that the Planck constant cannot be variable over space?

Answer

Take the example of a hydrogen atom.

The energy levels in the hydrogen atom are given by $$ E_n = -2\pi^2 \frac{m_e^4}{n^2 h^2}.$$ The spacing between two energy levels determines the frequency of an emitted photon when a radiative transition is made between them: $$ h \nu_{n_2\rightarrow n_1} = \frac{2\pi^2 m_e^4}{h^2}\left(\frac{1}{n_2^{2}} - \frac{1}{n_1^{2}}\right). $$

Thus the frequency of the transition will be proportional to $h^{-3}$. Thus if $h$ changes, then the frequency of spectral lines corresponding to atomic transitions would change considerably.

Now, when we look at distant galaxies we can identify spectral lines corresponding to the atomic transitions in hydrogen. As John Rennie says, these are redshifted and so this could be interpreted as a systematic spatial change in Planck's constant with distance from the Earth. However, this shift would have to be the same in all directions - since redshift appears to be very isotropic on large scales - and would thus place the Earth at the "centre of the universe", which historically has always turned out to be a very bad idea. Alternatively we could assume Planck's constant was the same everywhere is space but was changing with time - this would produce an isotropic signal. (NB: Cosmic redshift cannot be explained in this way since there are other phenomena, such as the time dilation of supernova light curves that do not depend on Planck's constant in the same way, that would be inconsistent).

That is not to say that these issues are not being considered. Most effort has been focused on seeing whether the fine structure constant $\alpha$ varies as we look back in time at distant galaxies. The reason for this is, as Count Iblis correctly (in my view) points out, the quest for a variable $h$ is futile, since any phenomenon we might try to measure is actually a function of the dimensionless $\alpha$, which is proportional to $e^2/hc$. We might claim that if either varied, it would produce a variation in $\alpha$, but this would amount to a choice of units and only the variation in $\alpha$ is fundamental.

So, in the example above, the hydrogen energy levels are actually proportional to $\alpha^2/n^2$. A change in $\alpha$ can be detected because the relativistic fine structure of spectral lines - i.e the separation in energy between lines in a doublet for instance, also depends on $\alpha^2$, but is different in atomic species with different atomic numbers.

There are continuing claims and counter claims (e.g. see Webb et al. 2011; Kraiselburd et al. 2013) that $\alpha$ variations of 1 part in a million or so may be present in high redshift quasars (corresponding to a fractional change of $\Delta \alpha/\alpha \sim 10^{-16}$ per year) but that the variation may have an angular dependence (i.e. a dependence on space as well as time).

electromagnetism - Using AC or DC to create an electromagnet

Does it matter if I use AC or DC as a source to create an electromagnet?

For example, suppose I have a coil which has about 50 turns. If I use the same AC or DC voltage, which will create a stronger magnetic field?

Answer

A DC circuit has current circulating in one direction.

An AC circuit has the current reversing direction and changing in amplitude with the frequency of AC.. The magnetic field will follow.

In a DC setup the magnetic field is constant and follows the right hand rule . This is what you should use if you want something like a permanent magnet or in order to increase the strength of a permanent magnet.

In an AC setup it is continually changing :

In alternating current (AC) electromagnets, used in transformers, inductors, and AC motors and generators, the magnetic field is constantly changing. This causes energy losses in their magnetic cores that are dissipated as heat in the core.

Use AC if you want a motor, for example.

As for stronger, DC is constant, while AC is alternating and falling in amplitude in the cycle, so DC will give the stronger field, except at the peak voltage of the AC cycle, when AC will be stronger. At the RMS voltage they will be equal but maybe of opposite sign, depending on the part of the ac cycle.

In addition with DC you can use a permanent magnet for a core and thus increase the value.

newtonian mechanics - Can Newton's laws of motion be proved (mathematically or analytically) or are they just axioms?

Today I was watching Professor Walter Lewin's lecture on Newton's laws of motion. While defining Newton's first, second and third law he asked "Can Newton's laws of motion be proved?" and according to him the answer was NO!

He said that these laws are in agreement with nature and experiments follow these laws whenever done. You will find that these laws are always obeyed (to an extent). You can certainly say that a ball moving with constant velocity on a frictionless surface will never stop unless you apply some force on it, yet you cannot prove it.

My question is that if Newton's laws of motion can't be proved then what about those proofs which we do in high school (see this, this)?

I tried to get the answer from previously asked question on this site but unfortunately none of the answers are what I am hoping to get. Finally, the question I'm asking is: Can Newton's laws of motion be proved?