particle physics - Pair-annihilation why does it occour?

Why does pair annihilation occur with particles and only their matching anti-particle? E.g., electrons and positrons, but not protons and positrons? What is the difference?

Answer

We (physicists) believe the reason is this: known symmetries and conservation laws. For example, the mutual annihilation of a proton and positron would remove $2\,e$ charge units from the Universe. This violates the conservation of charge principle, which can be seen to arise from the application of Noether's theorem to the global gauge symmetry of the classical electrodynamic Lagrangian, as discussed in QMechanic's answer to the question "Noether theorem and classical proof of electric charge conservation". Other similar conservation laws are conservation of color charges in QCD, conservation of lepton number, baryon number (quark number) and so forth. Your proposed reaction would also violate conservation of lepton number.

But ultimately the answer is simply experimental evidence, i.e. mutual annihilation simply doesn't happen experimentally unless between antiparticles. The failure to observe certain reactions is what led theoretical physicists to define Lagrangians with various continuous symmetries symmetries (and thus, through Noether's theorem, Noether charge (i.e. quantum number) conservations) as well as the notion of anti-particle itself in the first place as a way to organize our knowledge of experimental particle interaction observations. The theoretical reasons above are simply sufficient conditions that would explain our observations, but they are not necessary conditions. It's simply that, so far, our theories agree with experiment and we therefore have no reason to believe they are wrong.

thermodynamics - heat as a low quality energy

In a cyclic process, there is no change in internal energy. So the work done by the system must equal to the heat offered to the system. So if all heat is converted to work, how can heat by a low quality energy?

calculation puzzle - Number of days to fill the Lake with 1/32th part

There is a lake in which every day some new lotus flowers grow. Their growth follows a pattern; if on a given day there are X flowers, then on the next day there will be 2X flowers will be there. So everyday the number of flowers gets doubled.

Now the main question is:

If the whole lake gets covered with lotus flowers in 13 days, then how many days does it take to cover only 1/32th part of the lake?

Answer

Assuming you mean 1/32th:

It should take 8 days.

On day 9 you'll have 1/16th

On day 10 you'll have 1/8th
On day 11 you'll have 1/4th
On day 12 you'll have 1/2nd
On day 13 you'll have a full lake.

energy - $E=kT$ or $frac32kT$?

Basically, which is the correct formula for thermal energy, and is this the same as kinetic energy? My notes are pretty conflicting on this topic, and I'm getting pretty confused.

event horizon - Information encoded on the surface of a black hole

If an object that enters a black hole has its information content frozen at the event horizon, in what sense is it frozen? The usual analogy is of a hologram encoded in 2d which can be decoded into a 3d representation. The same analogy is used in describing the 3d universe we see as a representation of the holographic information encoded on the 2d "surface" of the universe.

1. 
   

   What is meant by "information content"? Or have I stated this incorrectly?

   
   


2. 
   

   The analogy leads one to wonder if there is a way to recreate the objects that have entered a black hole in any sense. Is the information frozen on the event horizon of a black hole detectable or visible in any real sense from the outside? How is one to imagine this effect intuitively?

   
   
   

I understand that in thinking about the holographic analogy I may have been going completely astray so please don't snipe at the question, simply correct me.

newtonian mechanics - Approaches to Sand on a Conveyor Belt

Say we are given a conveyor belt with sand falling onto it at rate $\Omega$. I am trying to find the power it takes for the belt to operate if it goes forward with constant velocity $v$, but using two different approaches I get two different answers.

The first way, I say that in one second, $\Omega$ amount of sand falls onto the belt, thus every second $\frac 1 2 \Omega v^2$ is required to continue moving the belt forward (simply by looking at kinetic energy difference) and $P=\frac 1 2 \Omega v^2$

The second way, I saw that $F=\frac {d(mv)} {dt}$, thus $F=\Omega v$. Finally, $P=Fv$ so $P=\Omega v^2$.

So which one is it: $P=\Omega v^2$ or $P=\frac {1} {2} \Omega v^2$, and where did I make an incorrect assumption in my two methods?

Answer

Your first answer gives you the rate at which the sand gains kinetic energy whereas your second answer gives you the rate at which work has to be done on the conveyor belt to keep it moving at constant speed.

When the sand falls on the belt in order to accelerate the sand to the same velocity as the belt there must be frictional forces between the sand and the belt.
There must also be relative movement between the belt and the sand during this acceleration phase as the sand cannot instantaneously to the velocity of the belt.
During the acceleration phase when there is slippage between the belt and the sand heat is generated and the rate of heat generation is the difference between your two answers.

So the motor which drives the belt increases the kinetic energy of the sand whilst heat is being generated due to the relative movement between the sand and the belt during the acceleration phase of the sand.

measurements - Why does the rule for multiplication/division take into consideration the no. of significant figures?

I've learned that the rule for multiplication, when taking into account the significant figures, is as follows:

The final result should retain as many significant figures as there are in the original number with the least significant figures.

So, by that definition, if I has the mass of an object = 4.237g and its volume = 2.51 cm^3, then the density would be 1.69 g cm^-3. However, let's take an example in which we have the side of a cube given to be 25.6 which has 3 significant figures. Now let's find out its volume: 16777.216. Now if I wanna report my result with the least significant figures (3), I would get an answer 168 which is plain wrong.

I could use scientific notation, but even there, I would have the significant figures which are significant.

Could someone help me out by explaining me this rule's implication? Thanks.

Answer

Now let's find out its volume: 16777.216. Now if I wanna report my result with the least significant figures (3), I would get an answer 168 which is plain wrong.

Your problem is that you seem to be truncating the value! When employing the significant figure rule, you turn all other values to zero. Trailing zeros are placeholders, so they can be eliminated from the reported value when using scientific notation. If you are not using scientific notation, you must record those zero's: $$ 16777.216 \Rightarrow 16800\neq168 $$ As compared to the scientific notation form, $$ 1.6777216\times10^4 \Rightarrow 1.6800000\times10^4\equiv1.68\times10^4 $$

classical mechanics - How to interpret instantaneous velocity using limit?

Mathematics is the language of physics therefore to understand physics is to understand its language. Based on definition of derivative the smaller the change of X of difference quotient the more it approach a certain value if it exist which we call the derivative. But what does it mean in physics? If derivative is the model use to describe instantaneous velocity then it means that the smaller you observe time interval experimentally, the ratio of distance over time approaches to a certain value. But why does it approach a certain value when we observe an object moves ? What causes it to approach a constant speed in smaller interval? and why call it the instantaneous velocity/speed? The concept of derivative only describe it but have no explanation.

astronomy - Does the Milky Way have dark matter satellite galaxies?

This recent paper by Weinberg et al. discusses that one potential problem with our current model of Cold Dark Matter (CDM) is that is predicts a greater number of satellite galaxies for the Milky Way than are actually observed, and satellite galaxies with larger masses than those the Milky Way has (although the paper does say that many of the Milky Way's satellite galaxies may be too dim for us to currently observe).

I know that at least dark matter galaxies are believed to have been observed (see this article). Is there any observational or other evidence for the existence of dark matter satellite galaxies to the Milky Way? And how have they been found?

Edit: As has been pointed out, this is a question we likely don't have the answer to yet (otherwise there wouldn't be recent papers posing the question still), so perhaps a better question would be how could/would you find dark matter satellite galaxies of the Milky Way?

Answer

This is actually a problem between simulations of structure formation and observations on a couple of different mass scales. Both galaxies and galaxy clusters appear to have nearly an order of magnitude more satellites than what is actually observed. The problem is dubbed the "missing satellite problem", or the Dwarf galaxy problem.

People have been asking the question, what if these dwarf galaxies are simply not massive enough to attract enough gas gravitationally for them to be visible? Some interesting ideas and work has been done to determine if these structures actually do exist (Yoon, Johnston, and Hogg - Clumpy Streams from Clumpy Halos: Detecting Missing Satellites with Cold Stellar Structures). Also, Beth Willman has done some interesting work on detecting the least luminous galaxies (dwarf galaxy companions to the Milky Way). In other words, it is quite possible that there are small collections of stars, dust, and gas inside these substructures that are just so faint, we can't find them unless we're looking closely.

I should also add that people have gone to lengths in the other direction, too. That is, to see how to alter the properties of dark matter in order to rid simulations of structures on that particular scale (see: Self-interacting dark matter).

The LCDM cosmological model is a very robust and well-supported model for our universe on scales of around ~1 Mpc and larger. Some discrepancies do exist, of which the missing satellite problem is but one of them.

special relativity - Lorentz-invariance of step function

I was reading about the Lorentz invariant integration measure $\int \frac{d^3k}{2E_K}$, and ways to prove that this was Lorentz invariant. Many of the proofs I have read use the step function (or Heaviside function) $\theta(k^0)$, and claim that this is obviously Lorentz-invariant (e.g Alex Nelson's notes). Other sources claim that there is some requirement that the Lorentz transformations have determinant = 1, or that $k^\mu$ is timelike. What are the actual conditions on the vector $k$ for the step function to be lorentz-invariant?

Answer

Proper, orthochronous Lorentz transformations can't change the sign of a $x^0$ component in a time-like four-vector - $dx^\mu dx_\mu$ and $\text{sign}\,x^0$ are both Lorentz invariants if $x_\mu$ is time-like. Because $\text{sign}\,x^0$ is Lorentz invariant, the step function $\theta(x^0)$ is Lorentz invariant - the condition is that $x$ is time-like.

A general Lorentz transformation that wasn't orthochronous could violate causality by making one observer see time-like events occurring in a different order.

logical deduction - Web of Spirits - a puzzle of romance

Spirit Web:

Can you figure out the interconnected web of affections among these characters using the provided clues?

Basic Rules:
1: There are exactly three existing relationships.
2: No character in a relationship is pursuing anyone.
3: Except as noted, all single characters are pursuing exactly 1 character.
4: There are no outside characters involved.

Characters and Clues:
Alloy (straight male): Pursued by exactly one character.
Aqua (straight female): Pursued by Fate; not pursued by anyone else.
Arc (gay male): Only character after Terra; not pursued by Emote.

Areo (straight male): Is being pursued by Frost, who is in turn pursued by Tick.
Emote (straight female): Pursuing someone no one else pursues; no one pursues Emote.
Fate (lesbian female): Pursuing two characters, one of whom pursues Alloy.
Force (bisexual female): In a relationship; no one is after her partner.
Frost (straight female): Pursued by two guys, neither of whom pursue Ivy.
Ivy (lesbian female): Pursued by two guys; pursuing one girl, who in turn pursues Fate.
Lum (straight female): Pursuing only one guy; pursued only by that guy.
Nick (straight male): In a relationship, not with Force or Volt; not pursued by anyone.
Null (straight male): Pursuing the same character as Ivy.
Psych (straight male): Not pursued by anyone.

Pyro (straight male): Pursuing two girls; pursued by two girls.
Shade (bisexual female): Single and pursued by two characters of differing genders.
Terra (bisexual male): Pursuing the same girl as Alloy.
Tick (straight male): Not pursued by anyone.
Tone (straight female): Pursuing two characters, one of whom pursues Ivy.
Volt (straight female): In a relationship; someone is after her boyfriend.

Character List:
Male: Alloy, Arc, Areo, Nick, Null, Psych, Pyro, Terra, Tick.
Female: Aqua, Emote, Fate, Force, Frost, Ivy, Lum, Shade, Tone, Volt.

Now, I am worried about two things:

1: Is the puzzle solvable with just these clues?
2: If so, are any of the clues unnecessary?

I'm fairly confident the puzzle is solvable, but I'm not quite sure.

Answer

I think I have it - I've come to no contradictions I can see, but a few people are left un-pursued. You didn't explicitly say that everyone was pursued by someone unless stated otherwise, so I assume it's OK to find people who aren't pursued.

Step 1:

Only Areo, Nick and Psych among the males are not known to pursue someone. Nick is in a relationship with someone who is not Force or Volt. Force’s partner must be male since the L/B females are all either pursuing someone else or being pursued by someone else, and “No one is after Force’s partner”. Likewise, Areo is discounted because someone is after him - so Force is partnered with Psych.

Step 2:

Volt is in a relationship (not with Nick). Someone is after her boyfriend. The only male not pursuing anyone and not in another relationship is Areo. He must be Volt’s boyfriend.

Step 3:

Emote is pursuing someone no one else pursues. This rules out Nick, Psych, and Tick (pursued by no one), Areo, Terra (pursued by others), and Pyro (known to be pursued by 2 girls). Alloy is pursued by someone Fate pursues; but Emote is pursued by no one, so she can’t be that person. She doesn’t pursue Arc per Arc's clue. This leaves Null.

Step 4:

The only female not pursuing anyone yet, in a relationship already, or explicitly stated as single is Aqua. She must be the female in relationship with Nick.

Step 5:

If Alloy and Terra are the two males after Ivy, only 1 person is after Terra, so Terra isn’t the one of Ivy’s pursuers who is pursued by Tone - that must be Alloy. Therefore Fate pursues Tone.

Step 6:

Null pursues someone who pursues Fate - that someone must be a L/B female. Of those: Fate won’t pursue herself; Force is in a relationship; Ivy is already pursued by two; so Null must be one of the two pursuing Shade. Ivy is then the other one pursuing Shade, by Null's clue.

Step 7:

All other males are pursuing only one person or in a relationship - so Lum must be pursuing and be pursued by Pyro.

Step 8:

Frost is being pursued by two people; Pyro is the only one left who could be her second pursuer. Pyro is being pursued by two females; Tone is the only one left who could be his second pursuer.

Answer:

Alloy: Pursues Ivy, Pursued by Tone

Arc: Pursues Terra, Pursued by no one
Areo: In a relationship with Volt, Pursued by Frost
Nick: In a relationship with Aqua, Pursued by no one
Null: Pursues Shade, Pursued by Emote
Psych: In a relationship with Force, Pursued by no one
Pyro: Pursues Frost and Lum, Pursued by Tone and Lum
Terra: Pursues Ivy, Pursued by Arc
Tick: Pursues Frost, Pursued by no one
Aqua: In a relationship with Nick, Pursued by Fate
Emote: Pursues Null, Pursued by no one

Fate: Pursues Aqua and Tone, Pursued by Shade
Force: In a relationship with Psych, Pursued by no one
Frost: Pursues Areo, Pursued by Tick and Pyro
Ivy: Pursues Shade, Pursued by Alloy and Terra
Lum: Pursues Pyro, Pursued by Pyro
Shade: Pursues Fate, Pursued by Ivy and Null
Tone: Pursues Alloy and Pyro, Pursued by no one
Volt: In a relationship with Areo, Pursued by no one

electromagnetism - Are there any real-world examples of refraction of light by magnetic permeability?

The question Fresnel Transmission Coefficient for Magnetic Field is interesting.

Thinking about it led me to reflect upon what little I know of the history of optics, with refraction by lenses and prisms being expressed in terms of an index of refraction, which at lower frequencies (microwaves) was related to the dielectric polarizability.

Today in optics texts it's usually the electric field amplitude rather than the magnetic field amplitude that's calculated, though we could just as well use either one with the proper conversions.

This led me to wonder Are there real-world examples of refraction due to magnetic permeability?

You can't focus light with an iron lens because it's opaque and possibly wouldn't have much permeability at such a high frequency. You might be able to make a microwave lens out of ferrite or some other low-loss medium, but for the purposes of this question only I won't call microwaves "light".

Question: In the wavelength range of IR (say about 10 microns or shorter) to near-UV (say about 100 nm or longer) are there any practical examples or demonstrations of refraction by the magnetic permeability of a material?

Could Quark model turn out to be false?

Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei.

Due to a phenomenon known as color confinement, quarks are never directly observed or found in isolation; they can be found only within baryons or mesons.

This sentence makes me very nervous: Due to a phenomenon known as color confinement

This sentence is Like I want to prove something spurious to save the subject (quark).

quantum mechanics - Have they really photographed light behaving both as a particle and a wave?

I just came across this article where they are claiming that they have photographed light behaving both as a wave and a particle!

The paper has been published in Nature Communications and I read the abstract which says,

Surface plasmon polaritons can confine electromagnetic fields in subwavelength spaces and are of interest for photonics, optical data storage devices and biosensing applications. In analogy to photons, they exhibit wave–particle duality, whose different aspects have recently been observed in separate tailored experiments. Here we demonstrate the ability of ultrafast transmission electron microscopy to simultaneously image both the spatial interference and the quantization of such confined plasmonic fields. Our experiments are accomplished by spatiotemporally overlapping electron and light pulses on a single nanowire suspended on a graphene film. The resulting energy exchange between single electrons and the quanta of the photoinduced near-field is imaged synchronously with its spatial interference pattern. This methodology enables the control and visualization of plasmonic fields at the nanoscale, providing a promising tool for understanding the fundamental properties of confined electromagnetic fields and the development of advanced photonic circuits.

They do not really talk about photographing light but instead they seem to talk about Surface plasmon polaritons. The Wikipedia article says that the "polariton" is a quasiparticle. However, they still claim to have photographed light behaving both as a wave and a particle.

Is this a valid interpretation of the experiment?

Answer

What they actually measured was not particle behavior. It was just a quantized energy transfer to the probing electrons. That corresponds to the absorption of individual photons, but it doesn't mean the Surface Plasmon Polariton (SPP) field was acting as a particle. It just interacted locally with the electron, as it must. Typically particle-like behavior happens when decoherence happens. In the two slit experiment, a superposition of a photon hitting the screen in two separate places is dynamically unstable and will decohere. Consequently, we only observe a single flash at a single point. Nothing like this happened in their experiment. They set up a standing wave in their SPP, and tossed electrons to image the interference fringes. If any interaction is to happen, it will happen at the electron's position, and if any decoherence is to take place in order to secure particle-like behavior, it will happen when the electron is detected. The SPP itself never acts like a particle -- it just acts like a quantum object, which, of course, it is. This paper would be much improved without mentions of confusing notions of wave particle duality. They have cool results as it is; anything else is spurious.

Here is a very good article explaining what it actually is, thereby clarifying the paper's findings.

Gravity - What happens when two objects of unequal masses fall freely towards the ground? (Revisited)

The common perception regarding what happens when two objects of equal sizes but unequal mass are allowed to fall freely towards the ground is that - both the objects make contact with the ground at the same instant.

This is attributed to the fact that the acceleration of both the objects towards the earth are the same for a given height. This was the reason for Galileo's expirement.

But the above fact ignores the fact that each of the three objects are attracting each other. In particular, the heavier object also attracts the lighter object, thus decreasing the distance between the lighter object and earth.

Note : I am ignoring the fact that the lighter object would also attract the heavier object (but to a lesser extent) and since the earth moves approximately in the same direction as a result of the attraction from the other two spheres, that is also ignored. Otherwise the earth can be said to move diagonally towards the heavier object (again to a small extent).

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I have included an illustration showing the movement of the lighter object towards the heavier one.

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As such the lighter object should fall first is it not?

Note : Generally, the two objects are placed close to each other (the distance-d between the objects is lesser than the height-h from the ground - 'd < h' ).

Continuing with the same logic, I have the following understanding :

When two spheres of equal size but unequal masses are allowed to fall freely towards the ground -

1.) From a height greater than the distance between the two spheres, the lighter sphere makes contact with the ground first.

2.) From same height as the distance between the two spheres, both the spheres make contact with the ground at the same time.

3.) From a height lesser than the distance between the two spheres, the heavier sphere makes contact with the ground first.

Note :

a.) The ground is also a sphere.

b.) The distance between the spheres are measured from their centers.

Is my understanding correct?

I also found this Phys.SE post. Which speaks of a scenario where in only two objects are involved. My question is similar to it.

PS: I have blogged about it before as well, saying it is so and also elaborated on two other cases ($d=h$ and $d>h$).

lateral thinking - What colour was the.... No, there's no damn bears here

Puzzlers of the world were gathering at the PuzzleMania conference-as a special event, celebrities from various fields of endeavour were invited to give their favourite puzzles.

It came to Ed the Explorer's turn:

This one will make you really think. Once, whilst exploring, me and my men headed for 100 units due south, then headed for 100 units due east and then headed for 100 units due north, and we ended up exactly where we had started. How is that possible?

But by then, there was a ripple going around the room. As soon as they had heard the first sentence, there were mutterings...

Oh not that one again-how many times have we heard this? Can't he come up with something original... Bo-ring..

But then Ed had them rather confused by adding on:

It has nothing to do with the North Pole... or poles of any sort... or anything to do with bears of any colour

So then where were Ed and his men and how did they manage to do this?

To address some ideas the puzzlers at the conference had, Ed would like to clarify:

1. They did actually travel the distances stated


2. The distances they traveled were in a straight line so when they went 100 units east, they ended up 100 units east of where they started and didn't just go around in circles.


3. When they had completed the 100 units north, they were back exactly where they started.

logical deduction - Escape the dungeon, by deciphering the codes and navigating the rooms to find a way out

You have been thrown into a dungeon with many rooms; you don't know who did this or why.

You start in room 37. You might be thinking "Why 37?", so I will explain this: "Why not?"

You need to find the way out of this place.

Of course, there is a lot of data that are encrypted in your way. Some of the data are really simple and easy to decode like "the first prime" or "0x30". Some of them are very hard, like "☏☀☑☀☂☇☔☓☄". All the data that should be decrypted by you in your journey are in bold text.

Most of rooms have exits to other rooms, however you will need to discover to which rooms. Note that if there is a way from room A to room B this DO NOT imply that there is a way from room B to room A. I.E. you should consider every passage as one-way only. A two-way passage is explicitly represented as a pair or two one-way only passages.

Post an answer explaining your movements step-by-step and room-by-room in order to successfully escape the dungeon. Do not forget to explain what items you collected on your journey and how you used them. Explain how you dealt with the trouble that you encountered. And of course, the most important: Explain how you decrypted all the boldtext secret data! I will accept the first answer that do all of this correctly, without gaps, errors or cheating.

Beware: I expect that this puzzle to be really hard, as in fact it is a series of puzzles in one.

And please, please. DO NOT FORGET TO POST YOUR ANSWERS IN SPOILER TAGS!

To put your answers in spoiler tags, you should add >! to the start of the lines inside your spoiler. To insert line breaks inside your spoiler tags, add a
in the end of the line.

^{Note: Feel free to edit this to fix possible spelling, grammar or punctuation errors, as long as it does not changes the bold text parts, except when obvious in clear english.}

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[Edited, long time after solved]: Made only somewhat spoilery rooms protected by spoilers. If you want to show them all, run this javascript: $(".postcell blockquote").removeClass("spoiler");. If you want to hide them all, run this other one: $(".postcell blockquote").addClass("spoiler");.

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Here are the rooms. Remember, you starts out in the 37:

Room 1:

That is the exit and the princess is already here just waiting for you!

Sure that the evil wizard was defeated, you and the princess walks out of the dungeon, going back to the castle.
And they lived happily ever after. ❤
The end!

Room 2:

You found a "☏☀☑☀☂☇☔☓☄". Strange, why the heck you would need this? You get it anyway.
Now go to the room numbered as the cube of this room.

Room 4:

A giant plutonium dragon comes from nowhere, put you in his mouth, crush your bones with his diamond jaw and eats you.
In his stomach, a boiling acid jet dissolves your body in a few seconds.
Game Over.

Room 5:

There is a man here. He said that he name is Caesar and he shares a secret to you. This secret is "kwws=22elw1o|24vSg{I}". What do this means?
There is a ventilation tube leading to the tri-hepta.
There is a strange device, called "sfuspqfmfu" but it does not works. If you manage to activate it somehow, it will lead you to room 11110.

There is window leading to room "how many legs there is in a chair?"
There is a big door leading to room "ruof-ytrihT".

Room 8:

There is a whirlpool in this room.
If you are unable to cope with it, you will be sucked through a hole direct to a room which the number divided by two is rounded to the number of this room.
If you can somehow stop it, you may grab a hook and crawl into the perfect room.

Room 10:

There is an arrow in this room, you decided to take it. There is something written in the arrow: "这是一个红色的鲱鱼 - Made in China".
There are three buttons: red, blue and yellow. You don't see any exit here, so what button do you press?

Room 12:

There is a door that leads to the Z room.
There are four pills here:
If you eat the red pill, you will be sent to the "naM hsaM" room.
If you eat the blue pill, you will be sent to the "ロコン" room.

If you eat the orange pill, you will be sent to the "Pidgeotto" room.
If you eat the white pill, you will be sent to the 21120110122101211011201112 room.
Warning: Pills do not respawn! So if you come back to this room later, the pills already taken won't be here again. I.E. you can't eat the same pill twice!
EDITED: If you take a pill, you need to eat it. You can't take all the pills at once and eat them somewhere else.

Room 13:

You found a "D". Now leave using "the same door that you used to come in".

Room 17:

The floor mysteriously disappears, leaving a dark bottomless pit where you instantaneously fall, forever!
Game Over.

Room 18:

There are three doors:
The left door goes to the iron room.
The front door to the room where you started.
In the ceiling, there is a big green steel locked door. If you are able to unlock it, it will lead you to the first prime room.

Room 21:

You found a "Красный ключ".
There are four tunnels here:
The left tunnel leads to room "square of the first prime".
The middle tunnel leads to the room with the pills.
The right tunnel leads to the 0x30 room.
The tunnel in the back leads to the room made by repeated numbers.

Room 23:

You found a magic bean!
There is a sign in this room: "◓●◒◒◙ ◔◈◉◓ ◉◓ ◁ ◒◅◄ ◈◅◒◒◉◎◇"
Two exits. The left one leads to room "@@@@@@@@". The right to room XLVIII.

Room 24:

There is a huge lion here. If you can't kill him quickly enough, you will become his dinner.
Behind the lion, should you be able to avoid him, there are two stairs:

The stairs going up leads to the room "Monica, Erica, Rita, Tina, Sandra, Mary, Jessica".
There is a massive red steel door which needs a red key. If you are able to open it, there is a stairs going down leading to the bad luck room.

Room 26:

You found a powerful gun. There is just one bullet there.
There is a window that leads to room "Butterfree".
There are two doors: The left one leads to room "neetneves" and the right one leads to the iron room.

Room 28:

WHOA! You found a big pot of gold!
Now, proceed to the room of that girls.

Room 30:

The front door leads to the room which the number is THE answer!
There is a window in the wall, but it is too small for you to pass through.
Looking through the window, you see some trees out there. There is a bullseye near one of the trees.
If you have the needed equipment (what is it?), you may shoot the bullseye.

Room 34:

You found the blue key!
There is a spring that may throw you away to the room "Mg". There is a closet that is a secret passage to the room which is a multiple of 11.

Room 37:

There is a blue box, but it is locked. You need to find a blue key to open it.
Two doors. The left door goes to room "Argon". The right goes to room "Micro Mike".

Room 40:

You found the cake!
This cake is so delicious and moist!
Now you can walk back to the previous room or go to the "Yt4$)" room.

Room 42:

There is a gnome in this room.

He said: "If you give me a pot of gold and a magic bean, I will make a rainbow! Everybody knows that there is a pot of gold in the end of the rainbow!"
Curious, you decide to ask for more information. So, he tells this:
"I used to be a prisoner in a dungeon just like you. But then, I took an arrow to the knee."
! Anyway, in this room there is a river called Tiber (why is the name of the river important?), leading to room X.

Room 44:

You found a glowing crystal key!
There are a window leading the room with the largest number.
There is an elevator leading to the first room with a number greater than this one.

Room 48:

A giant circular fast-spinning blade falls onto your head.
Then, your body is divided in two halves!
Guess what? You got a nice Game Over!

Room 56:

You found a black magic book and got it. Then you are teleported back to the starting room.

If you already have been in this room before and had already taken the book, you will be teleported to the room whose number is the first two-digit room number appearing in π.

Room 59:

You found something TERRA. Guess what it could be?
Now choose your way out: "dezessete" or "dezoito".

Room 63:

Congratulations, you won a million dollars!

Now you just need to leave this dungeon and you are a rich man!
There is an escalator going up here. It leads to the room with repeated numbers.
There is a door to the left it leads to the room 0x4.

Blue box:

You found the batteries for the "84 69 76 69 80 79 82 84 69 82"!

Shooting the bullseye:

You shoot through the window, and hit the bullseye just in the middle!
Immediately, a new trapdoor opens in the room 1A. This trapdoor leads to the room "五十六".

Red button:

You pressed the RED button:
BOOOOOM!!!! Game over!
You should know that this always happens when you press a red button!

Blue button:

You pressed the BLUE button:
An elephant comes out of nowhere and tramples you. Game over!

Yellow button:

You pressed the YELLOW button:
A secret passage opens in that "ending with '3s2 3p6'" room. It leads to room "◔◈◅ ◌●◒◄ ◉◓ ◍◙ ◓◈◅◐◅◒◄, ◉ ◌◁◃○ ◎●◔◈◉◎◇".
A trapdoor opens under you feet. You fell onto the room "NPPSTVPJWFSQFIUGPGMBITJSFCNVOFIUFTPIX".

If you decide to insert the arrow in your own knee:

Seriously? Are you crazy? What the hell are you thinking?
The arrow was poisoned. You became paralyzed. Your skin turns blue.
You fell to the ground in a deep pain and agony. You can't breath and your heart stops.
Congratulations! That was the most stupid thing that you could do in this game!
Guess what you got now? A giant, red, blinking and dancing "game over"! That is right dude, you lost the game and died for being so stupid!

Magic book:

In the book, it is written:

Abrakadabra!
. ορετόσσιρεπ ιχό ιεσάρδ αν ιεπέρπ αθ ςασ ήγρΟ ! άρεν ανέμγαρατ ατ όπα ωνάπ υομ ήμανύδ ητ ιαμύολακιπε , ςανώδιεσοΠ .

Giving the gold and the magic bean to the gnome:

The gnome receives the gold and the magic bean and a rainbow appeared right in front of your eyes, entering in the pot of gold.
You started to walk over it. That is incredible! You did not knew that you could walk over a rainbow!
You thanks the gnome and go away, walking over the rainbow into the blue sky.
The view here is so beautiful! You can see mountains, rivers and cities down there...
But then, you perceived something:

You leaved the dungeon! Congratulations! You are free!
And then, you remembered still another great thing:
Rainbows do not lasts for long time!
The rainbow disappears under your feet, and you fall, quickly going directly to the ground.
Oh no, you will face certain death! Except if...?

Answer

So here is my stab at this. I promise I did not look at any of Lopsy's work.

Room 2: ☏☀☑☀☂☇☔☓☄, Subtracting 9727 from the Unicode code points gives numbers, which correspond to the letters Parachute.
Room 2: the cube of this room, is 8.
Room 5: kwws=22elw1o|24vSg{I}, is a Caesar cypher. Subtracting 3 from each character gives http://bit.ly/1sPdxFz which shows a bunch of troll faces (red herring).
Room 5: tri-hepta, tri is for 3 and hepta for 7, so I'm gonna venture a guess and say that's 37. I've never used this door, though.
Room 5: sfuspqfmfu, decrementing each character by 1 and reversing the string spells teleporter.
Room 5: 11110, is binary for 30.
Room 5: How many legs there is in a chair?, I hope that's 4.
Room 5: ruof-ytrihT, that's 34 backwards.
Room 8: a room which the number divided by two is rounded to the number of this room, is 17 (could be 16 but that doesn't exist).
Room 8: perfect room. I'm very unsure about this one, but I'm gonna guess 21, because it's the perfect score in Black Jack. Edit: Now I have looked at Lopsy's solutions. ...facepalm... Perfect numbers were my first thought, but looking at the Wikipedia article I overlooked 28 and thought they went 6, 496, 8128... I'm gonna count this one as "solved correctly, but too blind". I've shortened the way out correspondingly.

Room 10: 这是一个红色的鲱鱼, mean This is a red herring in Chinese.
Room 12: Z, I believe this is just room 26 (alphabet and stuff).
Room 12: naM hsaM, "Mash Man" is game number 37 in Action 52.
Room 12: ロコン, is the Japanese name of Pokémon Vulpix, with Pokédex number 37.
Room 12: Pidgeotto, is also a Pokémon, with Pokédex number 17.
Room 12: 21120110122101211011201112, This is Morse code. 0 are separators, 1 short and 2 long signals. It spells XIPLUV which is VULPIX backwards. So this is another 37.
Room 13: D, that looks like a Bow to me!
Room 13: the same door that you used to come in, I think only 24 has a door leading to 13.
Room 18: Iron room, new guess: this is just 26 (the atomic number of Iron) in parallel to the Argon hint.
Room 18: the room where you started, is 37.

Room 18: the first prime, is 2.
Room 21: Красный ключ, is Russian for Red key (you have no idea how proud I was to be able to read that myself, despite my very rudimentary Russian lessons being some 6-10 years in the past :D).
Room 21: square of the first prime, is 4.
Room 21: room with the pills, is room 12.
Room 21: 0x30, is hexadecimal notation for 48.
Room 21: room made by repeated numbers, I believe "numbers" refers to the "digits" in the room number, so this would be 44.
Room 23: ◓●◒◒◙ ◔◈◉◓ ◉◓ ◁ ◒◅◄ ◈◅◒◒◉◎◇, Subtracting 9600 from each Unicode code point yields SORRY THIS IS A RED HERRING.
Room 23: @@@@@@@@, There's 8 of them, so that's just room 8.
Room 23: XLVIII, Roman numeral for 48.
Room 24: Monica, Erica, Rita, Tina, Sandra, Mary, Jessica, is from Mambo No. 5 (as a German, I didn't even have to look that up thanks to Lou Bega's cover^^).

Room 24: bad luck room, this has got to be 13, being the unlucky number.
Room 26: Butterfree, yet another Pokémon with Pokédex number 12.
Room 26: neetneves, is 17 backwards.
Room 26: iron room, so if my guess above is right, then this door leads to 26 itself.
Room 28: room of that girls, I think this refers back to the hint for Mambo No. 5.
Room 30: THE answer, should be the answer to life, the universe and everything: 42.
Room 34: Mg, is symbol for Magnesium, with atomic number 12.
Room 34: multiple of 11, that's 44.
Room 37: Argon, has atomic number 18.
Room 37: Micro Mike, another game from Action 52, with number 24.

Room 40: The cake is a lie. There's no way to get to this room, and therefore the Yt4$) likely doesn't mean anything at all.
Room 42: X, I thought this could be 24 as the 24th letter of the alphabet, or it could be 10 as a Roman numeral. I went with the latter, because I didn't find another way into room 10, and because the Tiber runs through Rome (then again, the alphabet in question is the Latin alphabet ;)).
Room 44: room with the largest number, is 63.
Room 44: first room with a number greater than this one, is 48.
Room 56: the starting room, is 37.
Room 56: the room whose number is the first two-digit room number appearing in π, is 59.
Room 59: TERRA, is a ROT-13 cypher of "GREEN" so this is the Green key.
Room 59: dezessete, Portuguese for 17.
Room 59: dezoito, Portuguese for 18.
Room 63: room with repeated numbers, again, I believe this refers to the digits, so that's 44.

Room 63: 0x4, should be hexadecimal notation, which is just 4.
Blue box: 84 69 76 69 80 79 82 84 69 82, ASCII codes of TELEPORTER.
Bullseye: 1A, I'm hoping is also hexadecimal, which would be 26.
Bullseye: 五十六, Japanese (I think?) for 56.
Yellow button: ending with '3s2 3p6', could refer to the electron configuration of Argon, which would be 18 again.
Yellow button: ◔◈◅ ◌●◒◄ ◉◓ ◍◙ ◓◈◅◐◅◒◄, ◉ ◌◁◃○ ◎●◔◈◉◎◇, subtracting 9600 from the Unicode code points gives THE LORD IS MY SHEPERD, I LACK NOTHING which is in Psalm 23.
Yellow button: NPPSTVPJWFSQFIUGPGMBITJSFCNVOFIUFTPIX, decrement each letter by 1 and reverse the string to get WHOSETHENUMBERISHALFOFTHEPREVIOUSROOM. Since the button is in room 10, and (I think) the only way to get there is from 42, this is 21.
Magic book: . ορετόσσιρεπ ιχό ιεσάρδ αν ιεπέρπ αθ ςασ ήγρΟ ! άρεν ανέμγαρατ ατ όπα ωνάπ υομ ήμανύδ ητ ιαμύολακιπε , ςανώδιεσοΠ . Reverse this to get actual Greek. It translates to Neptune, I invoke my power over troubled waters! Rage you should act no more., so the book can be used to tame the whirlpool. I swear this is a reference to The Tempest.
Rainbow: Except if...?, you've got a parachute...

Phew... that's out of the way. Let's get out of this dungeon!

First, I think 44 is a dead end, because it leads to 48 (instant death), and 63, which in turn leads only back to 44 or to 4 (instant death). Furthermore, you don't need either glowing crystal key, nor any money. Likewise, 1 is probably a red herring, I don't think there's a way to get to it. Furthermore, there's no evil wizard, and the puzzle doesn't require me to rescue a princess. 40 also cannot be reached.

Let's go!

37 -> Go to 18
18 -> Go to 26
26 -> Take gun, go to 26. (because I can!)
26 -> Go to 26. (because I can!)
26 -> Go to 26. (because I can!)
26 -> Go to 12.
12 -> Red pill to 37.

37 -> Go to 24.
24 -> Kill lion with gun, go to 5
5 -> Go to 34
34 -> Take blue key, go to 12
12 -> Blue pill to 37.
37 -> Unlock blue box, take batteries, go to 24
24 -> Go to 5.
5 -> Plug batteries in teleporter, go to 30
30 -> Go to 42
42 -> Go to 10

10 -> Take arrow, press yellow (opens door in 18 to 23), go to 21
21 -> Take red key, go to 12
12 -> White pill to 37
37 -> Go to 24
24 -> Unlock red door, go to 13
13 -> Take the bow, go to 24
24 -> Go to 5
5 -> Use teleporter to go to 30
30 -> Shoot bullseye with bow and arrow, to open trapdoor in 26 to 56, go to 42
42 -> Go to 10

10 -> Go to 21
21 -> Go to 12
12 -> Door to 26
26 -> Use trapdoor to 56
56 -> Take Magic Book, go to 37
37 -> Go to 18
18 -> Secret passage to 23.
23 -> Take magic bean, go to 8.
8 -> Use Magic Book, go to 28
28 -> Take pot of gold, go to 5

5 -> Go to 34
34 -> Go to 12
12 -> Door to 26
26 -> Trapdoor to 56
56 -> This time go to 59
59 -> Take green key, go to 18
18 -> Unlock green door, go to 2
2 -> Take Parachute, go to 8
8 -> Use Magic Book, go to 28
28 -> Go to 5

5 -> Use teleporter to go to 30
30 -> Go to 42
42 -> Give gold to gnome
Rainbow -> Use parachute to survive the fall and land somewhere safely outside the dungeon!

Woohoo!

I believe there's a shorter way through the dungeon (especially the second half), but this one does the trick. Thanks to Victor for a couple of hints when I was making things more complicated then they were. :)

hamiltonian formalism - Conditions for periodic motions in classical mechanics

What are the conditions for classical motion to be periodic? In one dimension, if the motion is bounded, then it is also periodic. However, I don't think this generalizes to higher dimensions.

I am interested in some general condition that does not require obtaining the exact solution. This would be useful, for example, in applying the action-angle coordinate method, which does not require solving the equations of motions, but assumes that the motion is periodic in the first place.

Answer

The sufficient, but not necessary condition for a system to present regular motion is that it's integrable. An integrable system has enough constants of motion to restrict the system orbits in phase space $-$ specifically, its phase space is foliated by invariant regular tori. Regular means non-chaotic, i.e., periodic or quasi-periodic motion. As for how to find constants of motion, there are a couple of methods, in particular using Poisson brackets (see also this answer).

Now, there can be plenty of periodic motion in non-integrable systems. In particular, as kakaz pointed out in their comment, the KAM theorem is very relevant here: it states, loosely speaking, that the periodic orbits of a perturbed integrable system will remain regular, as long as they do not resonate with the perturbation and the latter is not too strong, which means that large fractions of the perturbed phase space typically remain regular under small perturbations.

Note that 2D systems are also regular (if they are smooth and otherwise well behaved), and that the question Are there necessary and sufficient conditions for ergodicity? is in some sense a complement of the present question.

logical deduction - Taming a Bomb Puzzle (Part 1 of 3)

Ampera Bridge, the famous landmark of Palembang City is in danger! A terrorist has sticked a bomb in the middle of Ampera Bridge. The bomb cannot be removed and it can explode anytime.

The bomb has N buttons, labeled from 1 to N (inclusive). It will explode right after pushing any of those buttons T times except one thing as follows.

Among the buttons, there is exactly one button, namely X. The bomb is designed so that when X has been pushed, it will sound “BEEP” but delayed at next K button pushings after it was pushed. In other word, if X was pushed at the i-th push, its “BEEP” can be heard at the (i+K)-th push. You don't know the value of K, but K is guaranteed to be between 0 to N-1. Of course, you also don't know the value of X.

Whenever the "BEEP" has been heard for N times (not necessarily consecutive), the bomb can be deactivated (tamed), of course, as long as the total number of button pushes does not exceed T times. When you hear the "BEEP", you may not know from which push it is made. But surely, if at the i-th push you hear the "BEEP", then your (i-K)-th push must be X.

To help you follow the definition and the example, here's a quick glossary:

  N: the number of buttons
  T: at this many button presses, the bomb goes off
  X: the button you need to identify
  K: an unknown but constant delay before the beep, measured in button presses. From 0 up to, but not including N
  i: the total press count, when a particular button was pressed

You, as a top bomb tamer, are to find a sequence of button pushes for deactivating the bomb.

---

Let's say you have the information of N = 4 and T = 20.

You have no clue at the beginning that this bomb has a value of X = 3 and K = 2.

 -----------------------
| No | You Push | Beep? |  
 -----------------------
|  1 |     3    |   -   |

|  2 |     2    |   -   | 
|  3 |     3    | BEEP! | 
|  4 |     4    |   -   | 
|  5 |     1    | BEEP! | 
|  6 |     4    |   -   | 
|  7 |     3    |   -   | 
|  8 |     1    |   -   | 
|  9 |     1    | BEEP! | 
| 10 |     1    |   -   | 
| 11 |     1    |   -   | 

| 12 |     3    |   -   |
| 13 |     3    |   -   | 
| 14 |     3    | BEEP! |

Some key points:

• You push button X = 3 as the first push, but its "BEEP" is heard after third (1+K = 1+2 = 3) push. Same as the third push's "BEEP" is heard at fifth push.


• The bomb hasn't been tamed at 12-th push. It will be tamed after 14-th push, when the N = 4-th "BEEP" is heard.


• If T is 13 or less, the bomb will explode in this case.

---

Let's say, you get the information about the value of K from somewhere (maybe the terrorist wrote it down on the back of the bomb).

The bomb has N = 50 buttons. What is the least possible value of T which still guarantees you to tame the bomb?

Bonus: Can you generalize T with any N?

Note that K can be 0 to 49 (N-1).

---

_{^{This puzzle is based on a competitive programming problem authored by me. It is used in Indonesia National Science Olympiad in Informatics 2016. The link is here (spoiler ahead for further parts!)}}

Answer

Building on Lolgast's answer, it's actually possible to do it in a few presses less! To be exact, it's

N + N-1 + N-(M+1) + N-1, or 4N - M - 3
With M being the largest number where (1+2+..+M <= N-1).

So at the very least, in order for the bomb to always be defusable, T must be

4N - M - 2

Explanation:

Let K = N - 1,

There are generally 4 'phases' to the presses:

1) The 'initial check' phase - just press every button in order, 1 to N. if there are no beeps by the time you press N, 1 is no longer a possible correct button. This costs N presses

2) The 'wait' phase - we know K = N-1, so at worst case (last button is the correct button) we'll know the correct button at the latest in N - 1 steps.

3) The 'press' phase - Press the correct button as many times as needed to reach N times. This costs N - [however many times we've pressed the correct button earlier]

4) The 'second wait' phase - wait for time to pass until there are N beeps. This is N-1 from the last press.

Lolgast's answer assumes we just repress every button from the start for the second phase, so at worst case (last button is correct), we would have pressed it twice, for the total of 4N - 4.

But see, if (2) is the correct button, then phase 2 will only last 1 step. Generalizing that, if (X) is the correct button, then phase 2 will last X-1 steps. On the other hand, phase 3 can be shorter depending on how many times you press (X) in phase 2 as well!

The key here is to make sure the free time we have in phase 2 is not wasted - for larger X, make sure we've pressed it as many times as possible to minimize the length of phase 3.

It's a bit difficult to explain the idea by words, so here's a visualization of the first 2 phases with 7 buttons (- will separate phase 1 and 2):

Instead of
1 2 3 4 5 6 7 - 2 3 4 5 6 7
Worst case, If 7 is the right button, it has been pressed 2 times by 2N-1 steps.
So total of Phase 2 + 3 would be N-1 + N-2 = 2N-3.

We can do
1 2 3 4 5 6 7 - 5 6 6 7 7 7
If 7 is the right button, it has been pressed 4 times by 2N-1 steps, or
If 6 is the right button, it has been pressed 3 times by 2N-2 steps, or
If 5 is the right button, it has been pressed 2 times by 2N-3 steps, and so on
This results in a total of Phase 2 + 3 to be N + N - 5 = 2N-5.

Using this strategy, the minimum number of presses required is:
N + N-1 + N-(M+1) + N-1, or 4N - M - 3

With M being the largest number where (1+2+..+M <= N-1).

mathematical physics - A book on quantum mechanics supported by the high-level mathematics

I'm interested in quantum mechanics book that uses high level mathematics (not only the usual functional analysis and the theory of generalised functions but the theory of pseudodifferential operators etc, certainly the modern mathematics). If there isn't something similar please give me a reference to the book that is strictly supported by mathematics (given a set of mathematically descripted axioms author develops the theory using mathematics as a main tool).

Answer

1.

• E. Zeidler, Quantum Field theory I Basics in Mathematics and Physics, Springer 2006. http://www.mis.mpg.de/ezeidler/qft.html
  
  

  is a book I highly recommend. It is the first volume of a sequence, of which not all volumes have been published yet. This volume gives an overview over the main mathematical techniques used in quantum physics, in a way that you cannot find anywhere else.

  
  

  It is a mix of rigorous mathematics and intuitive explanation, and tries to build "A bridge between mathematicians and physicists", as the subtitle says. It makes very interesting reading if you know already enough math and physics. You need a thorough knowledge of classical analysis, and some acquaintance with differential geometry and functional analysis. Apart from that, the book gives references to additional reading - plenty of references as entry points to the literature for topics on which your background is meager.

  
  

  As regards to your request for high level mathematics (in the specific form of pseudo-differential operators, etc.), Zeidler discusses - as Section 12.5 - on 28 (of 958 total) pages microlocal analysis and its use, though there is only two pages specifically devoted to PDO (p.728-729), but he says there (and emphasizes) that ''Fourier integral operators play a fundamental role in quantum field theory for describing the propagation of physical effects'' - so you can expect that they play a more prominent role in the volumes to come.

  
  

  But, of course, PDO are implicit in all serious high level mathematical work on quantum mechanics even without mentioning them explicitly, as for example the Hamiltonian in the interaction representation, $H_{int}=e^{-itH_0}He^{itH_0}$, is a PDO. Work on Wigner transforms is work on PDOs, etc..

  
  

  2.

  
  

  Other books using PDO, much more specialized:

  
  


• G. B. Folland, Harmonic analysis in phase space


• A.L. Carey, Motives, quantum field theory, and pseudodifferential operators


• A. Juengel, Transport equations for semiconductors



• C. Cercignani and E. Gabetta, Transport phenomena and kinetic theory


• N.P. Landsman, Mathematical topics between classical and quantum mechanics


• M. de Gosson, Symplectic geometry and quantum mechanics


• P. Zhang, Wigner measure and semiclassical limits of nonlinear Schroedinger equations
  

  3.

  
  

  Finally, as an example of a book that "is strictly supported by mathematics (given a set of mathematically described axioms, the author develops the theory using mathematics as a main tool)", I can offer my own book

  
  


• A. Neumaier and D. Westra, Classical and Quantum Mechanics via Lie algebras.

general relativity - Is a singularity a real thing?

I've heard the work a few times now, the most recent in the star trek film. Is a singularity a real thing? If so what is it?

Answer

The word "singularity" is generally used to denote that some quantity becomes infinite or in general becomes undefined (i.e. cannot be expressed as a finite real number).

One particularly common use is in general relativity. At the event horizon of a black hole (the surface of no return), often coordinates that were well behaved far away run into trouble. For example, the time as measured by a clock infinitely far away slows down and comes to a halt near that surface, so far-away observers see stuff falling into a black hole slower and slower, never quite making it.

However, the above is only what is known as a coordinate singularity - something went wrong with your coordinate system, but nothing physically "broke." For a more benign example, the North Pole of the Earth has a coordinate singularity if you use standard longitude and latitude. What is the longitude of the North Pole? Well, it's undefined. There is nothing physically important about this statement - it merely reflects the choice of a poor coordinate system for that region. (In fact it can be shown mathematically that no sphere can be covered entirely by a singularity-free coordinate system.)

There can also be "true" singularities. In general relativity you might compute some coordinate-independent quantity - the (scalar) curvature of spacetime for instance - and see that it goes to infinity. No change of coordinates will "fix" this, and it means physically meaningful quantities are behaving badly. In classical general relativity, it has been proposed (see the cosmic censorship hypothesis) that any such true singularities (as might exist inside the event horizons of black holes) are always shielded by event horizons, so information about them can never reach us. In models where this is true, it doesn't even make much sense to talk about the existence of such things, since their properties by definition can never be studied.

One can also cast singularities in terms of geodesics - the paths that objects, light, and anything else follow in spacetime when no other forces act on them. A singularity is a point where a geodesic just ends. You reach that point and then that's it - the laws of physics have nothing more to say about your position.

In these latter two cases, if the singularities are "real" and somehow not hidden from us, they signify either (1) the universe has stopped being well defined, and therefore all bets are off when it comes to predictability and science in general, or (2) our theory is incomplete and needs more work. Most people believe in choice (2).

general relativity - How can anything ever fall into a black hole as seen from an outside observer?

The event horizon of a black hole is where gravity is such that not even light can escape. This is also the point I understand that according to Einstein time dilation will be infinite for a far-away-observer.

If this is the case how can anything ever fall into a black hole. In my thought experiment I am in a spaceship with a powerful telescope that can detect light at a wide range of wavelengths. I have it focused on the black hole and watch as a large rock approaches the event horizon.

Am I correct in saying that from my far-away-position the rock would freeze outside the event horizon and would never pass it? If this is the case how can a black hole ever consume any material, let alone grow to millions of solar masses. If I was able to train the telescope onto the black hole for millions of years would I still see the rock at the edge of the event horizon?

I am getting ready for the response of the object would slowly fade. Why would it slowly fade and if it would how long would this fading take? If it is going to red shift at some point would the red shifting not slow down to a standstill? This question has been bugging me for years!

OK - just an edit based on responses so far. Again, please keep thinking from an observers point of view. If observers see objects slowly fade and slowly disappear as they approach the event horizon would that mean that over time the event horizon would be "lumpy" with objects invisible, but not passed through? We should be able to detect the "lumpiness" should we not through?

Answer

Indeed, nothing can get under the horizon. The stuff close to the event horizon does move outwards as the BH radius increases. Even more with any BH deformations such as waves on its surface, the tidal deformations or the change of the rotation speed, all the oblects close enough to the horizon remain "sticked" to it and follow all the changes of the BH form. All objects close enough to a rotating BH horizon, rotate with it at the same speed. If a BH moves, so does everything close enough to its surface, including the things located on the side of the direction of the move. If anyone interested what mechanism make such sticking possible, it is called frame-dragging.

You may ask then, how a black hole can appear then and the horizon form. It is conjectured that they cannot, and the only possible black holes are the hypothetical primordial black holes that existed from the very beginning of the universe.

The objects that can be very similar to black holes are called collapsars. They are virtually indistinguishable from actual black holes after a very short time of the formation. They consist only of matter outside the radius of the event horizon of a BH with the same mass. This matter is virtually frozen on the surface like with actual BH, due to high gravity level.

Such collapsars possibly can become BHs for a short time due to quantum fluctuations and thus emit hawking radiation.

Astrophysicists do not separate such collapsars from actual black holes and call all them BHs due to practical reasons because of their actual indistinguishability.

Here is a quote from one paper that supports such point of view:

Our primary result, that no event horizon forms in gravitational collapse as seen by an asymptotic observer is suggestive of the possibility of using the number of local event horizons to classify and divide Hilbert space into superselection sectors, labeled by the number of local event horizons. Our result suggests that no operator could increase the number of event horizons, but the possibility of reducing the number of pre-existing primordial event horizons is not so clear and would require that Hawking radiation not cause any primordial black hole event horizons to evaporate completely.

source

astrometrics - Reverse Sun position algorithm?

To find the sun's position (elevation, azimuth) we can use many algorithms including the PSA algorithm.

These algorithm share the fact that the input is the local time, longitude and latitude and the output is the elevation and azimuth.

is there any algorithms that we can use to reverse this, the input would be the sun position and the output would be the local time ?

thanks

standard model - CKM matrix - Unitarity triangles

I currently read an article and they mentioned, that using the unitarity relation between the 1. and 3. column of the CKM matrix, one can easily show that the area spanned of the unitarity triangle is given by $2A = |Im(V_{ub}V^*_{ud}V^*_{cb}V_{cd})|$. So I tried to show that by myself. I have absolutely no idea if this is gonna work, but here is my attempt:

I started with the unitarity relation of the 1. and 3. column:

$V^*_{ub}V_{ud} + V^*_{cb}V_{cd} + V^*_{tb}V_{td} = 0$

This relation can be represented in the complex plain as a triangle, called unitarity trianlge. I made then the following to vectors:

$\vec{a} = \begin{pmatrix}Re(V^*_{ub}V_{ud})\\Im(V^*_{ub}V_{ud}) \\ 0\end{pmatrix}, \vec{b} = \begin{pmatrix}Re(V^*_{cb}V_{cd})\\Im(V^*_{cb}V_{cd}) \\ 0\end{pmatrix}$

Then taking the cross product of these two vectors should yield the mentioned area in the article:

$2A=|\vec{a}\times \vec{b}|= |\begin{pmatrix}0\\ 0 \\ Re(V^*_{ub}V_{ud})Im(V^*_{cb}V_{cd})-Im(V^*_{ub}V_{ud})Re(V^*_{cb}V_{cd})\end{pmatrix}|= |Re(V^*_{ub}V_{ud})Im(V^*_{cb}V_{cd})-Im(V^*_{ub}V_{ud})Re(V^*_{cb}V_{cd})|$

Now I dont see how that equals $|Im(V_{ub}V^*_{ud}V^*_{cb}V_{cd})|$?

Thank you.

Answer

You use that Im$(ab)= $Re$(a)$Im$(b) + $ Re$(b)$Im$(a)$, for $$a = V_{ub}V^*_{ud}\qquad\qquad b=V^*_{cb}V_{cd}$$ to get

\begin{align*} \text{Im}(V_{ub}V^*_{ud}V^*_{cb}V_{cd})&=\text{Re}(V_{ub}V^*_{ud})\text{Im}(V^*_{cb}V_{cd}) + \text{Re}(V^*_{cb}V_{cd})\text{Im}(V_{ub}V^*_{ud}) \\ &=\text{Re}(V_{ub}^*V_{ud})\text{Im}(V^*_{cb}V_{cd}) - \text{Re}(V^*_{cb}V_{cd})\text{Im}(V_{ub}^*V_{ud}) \end{align*} Where in the second equation I used \begin{align*} \text{Re}(a^*) = \text{Re}(a) \qquad\qquad \text{Im}(a^*) = -\text{Im}(a) \end{align*}

general relativity - How does a mass create the gravitational field of GR?

As far as I understand, and please correct me if I am wrong, but the basic idea of general relativity is that spacetime is curved by matter. What we call gravity is then not a force as per Newton but a consequence of the geometry of space time. It is assumed that there is a gravitational field produced by a mass which bends space time.

So my question is; How does a mass create the gravitational field?

electromagnetism - How is a spherical electromagnetic wave emitted from an antenna described in terms of photons?

When an atenna transmits radiowaves isn't it true that the electromagnetic pulse is radiated away from the accelerating electron as a spherical wave in all directions simultaneously, and if so how can the associated photon be "everywhere" on this rapidly expanding sphere?

special relativity - Inertia on relativistic mass when particle is near speed of light

Inertia is directly proportional to mass but what happens when something travel to speed near to light. Its relativistic mass tends to infinity but that is false mass so I want to know if inertia is applied to relativistic mass. Because if we infinite the mass then the inertia would also be infinite then it will never reach speed of light.

The answer to this riddle has a hole in the middle

Here's an easy riddle I got from a book:

The answer to this riddle has a hole in the middle,
And some have been known to fall in it.

In tennis, it's nothing, but it can be recieved,
And sometimes a person may win it.
Though it's not seen or heard it may yet be percieved,
Like princes or bees it's in clover.
The answer to this riddle has a hole in the middle,
And without it, one cannot start over.

Answer

Heres my guess:

Answer:

Love

Reasoning: The answer to this riddle has a hole in the middle,

the letter o is a hole and in the middle (kind of) of the word love

And some have been known to fall in it.

You can fall in love

In tennis, it's nothing, but it can be recieved,

Love in tennis is 0

And sometimes a person may win it.

You can woo someone and "win their love" from @Matt

Though it's not seen or heard it may yet be percieved,

Not a physical object but can be felt

Like princes or bees it's in clover.

Princes love to be in clover (rich) and bees love to be in clovers (plant)

The answer to this riddle has a hole in the middle,

Same as first line

And without it, one cannot start over.

Over starts with o, which is in the middle of love

knowledge - Crossword gone overboard

_{This puzzle is part 24 of Gladys' journey across the globe. Each part can be solved independently. Nevertheless, if you are new to the series, feel free to start at the beginning: Introducing Gladys.}

---

Dear Puzzling,

By the time you receive this message, I may have already returned from my trip. Today I visited a luxurious mansion built by an important person. I hope your board game skills are not rusty, because you'll need them for today's puzzle!

Wish you were here!
Love, Gladys.

Across
1. Large inlet
5. Local name for Scotland
9. More delicate
10. A place where a fly is a nuisance
11. An African country or its capital city
15. Less feral

16. A Uruk-hai, for example
18. Native of an Arab state
19. Zilch
20. Precious stone
21. Card below trey
22. Evaluated

Down
1. Perfect is the enemy of —
2. Prefix for -cycle or -brow
3. Short-range network

4. — as a fiddle
5. Arsenals
6. Instead of = in — of
7. Grandmaster Larsen
8. Newspaper piece
12. Outlaw assassinated by coward
13. Members of the clergy
14. Arizona landmark: Horseshoe —
15. Ancient Roman attire
16. Legal obligation

17. Sushi ingredient

---

_{Gladys will return in "Thousands and thousands of words".}

Answer

Once again, I feel like I’m swooping in and snatching the answer, using the excellent solutions already provided by @Glorfindel for the chess puzzles and @Jay for the crossword. Thanks to you both!

If we:

Only use white’s moves for the chess solutions per @Glorfindel’s answer, and then map them accordingly according to @Jay’s crossword, we get Tjong a Fie which is a luxurious mansion in North Sumatra, Indonesia.

Specifically:

a4: T
b5: J
a6: O
b7: N
a8: G
—
c7: A
—

d8: F
f7: I
g7: E

electrostatics - Field between the plates of a parallel plate capacitor using Gauss's Law

Consider the following parallel plate capacitor made of two plates with equal area $A$ and equal surface charge density $\sigma$:

The electric field due to the positive plate is

$$\frac{\sigma}{\epsilon_0}$$

And the magnitude of the electric field due to the negative plate is the same. These fields will add in between the capacitor giving a net field of:

$$2\frac{\sigma}{\epsilon_0}$$

If we try getting the resultant field using Gauss's Law, enclosing the plate in a Gaussian surface as shown, there is flux only through the face parallel to the positive plate and outside it (since the other face is in the conductor and the electric field skims all other faces).

$$\Phi = \oint \vec{E}\cdot\vec{dA} = EA$$

where $E$ is the electric field between the capacitor plates. From Gauss's Law this is equal to the charge $Q$ on the plates divided by $\epsilon_0$

$$\frac{Q}{\epsilon_0}\implies E = \frac{Q}{A\epsilon_0} = \frac{\sigma}{\epsilon_0}$$

I know there is something fundamentally incorrect in my assumptions or understanding, because I frequently get conflicting results when calculating electric fields using Gauss's Law. I am, however, unsuccessful in identifying this.

Edit: Also, another problem I noticed was that even if we remove the negative plate from the capacitor and then apply Gauss's Law in the same manner, the field still comes out to be $\sigma/\epsilon_0$ which is clearly wrong since the negative plate contributes to the field. So, maybe the problem is in the application of Gauss's Law.

statistical mechanics - Deriving Boltzmann statistics from the maximum entropy principle

In some lecture notes I have, the author derives the expectation value of the occupation numbers for a discrete system of fermions as follows:

Consider all states that have a certain energy $\varepsilon_s$. There shall be $a_s$ such states, and $n_s$ particles occupying these states. Then, the number of configurations in this energy level is $$ W_s = \frac{a_s!}{n_s!\left(a_s-n_s\right)!}$$

If one now considers all energy levels, the total number of possible states for the whole system is $$ W = \prod_s W_s $$ where $s$ enumerates the energy levels. (I think countably infinitely many levels should not be a problem, would they?)

Now, one can try and maximize the entropy $S=k\ln\left(W\right)$ under the constraints that the particle number is fixed, $N=\sum_s n_s$, as is the total energy, $E=\sum_s n_s\varepsilon_s$. Introducing the Lagrangian multipliers $\alpha$ and $\beta$ in $$ \Lambda := \frac{S}{k} - \alpha \left(\sum_s n_s - N\right) - \beta \left(\sum_s n_s \varepsilon_s - E\right) $$ one indeed finds an extremum for $\Lambda$ for $$ n_i = \frac{a_i}{1+\exp\left(\alpha+\beta\varepsilon_i\right)} $$ which is, up to a factor, the Fermi-Dirac statistic once the Lagrangian multipliers are identified to be $\alpha = - \frac{\mu}{k_B T}$ and $\beta=\frac{1}{k_B T}$.

Now, in a side remark, the lecture notes claim that, if one had assumed a classical system where $W = a_s^{n_s}$, one would have obtained Boltzmann statistics: $n_s = \exp\left(-\alpha-\beta \epsilon_s\right)$.

I assume that instead of $W$, the author meant to write $W_s$. From the form of $W_s$, I conclude that the system under consideration has discrete energy levels, and that the particles do not obey the Pauli principle and are distinguishable from one another. In these circumstances, $a_s^{n_s}$ seems to give the right number of configurations within the energy level $\epsilon_s$, the total number of configurations again being $W=\prod_s W_s$.

However, $S$ now is linear in the $n_s$, so that differentiation of the new $\Lambda$ with regard to some $n_i$ gives an expression independent of any of the $n_k$.

What went wrong? Why did the procedure seem to work in the first case, but not in this? Or did I make a (conceptual?) mistake somewhere along the line?

Any input would be greatly appreciated!

Answer

You are missing the term $\frac{1}{n_s!}$ from the product, i.e. $$W = \prod_s \frac{a^{n_s}}{n_s!}$$ from which the wanted result follows.

Obviously in a quantum sense classical particles are distinguishable, so the term does not immediately arise from indistinguishability per se. Rather, the reason the "extra" term appears is because we ought to be looking at the macroscopic realizations of the system, and it does not matter for the macroscopic state if we switch two particles, and it is in this sense that they are indeed indistinguishable.

quantum interpretations - How do we know particles exist? Aren't they just waves?

In the book "A Briefer History of Time" Stephen Hawking wrote:

The unpredictable, random element comes in only when we try to interpret the wave in terms of the positions and velocities of particles. But maybe that is our mistake: maybe there are no particle positions and velocities, but only waves. It is just that we try to fit the waves to our preconceived ideas of positions and velocities. The resulting mismatch is the cause of the apparent unpredictability.

Are there evidences that disprove this hypothesis?

If true, would it eliminate most of the apparent quantum paradoxes, and necessity to "Shut up and calculate!" for those who attempt to interpret quantum physics with common sense?

Edit: I assume that S. Hawking is aware of Standard Model, and he considers this statement as a legitimate hypothesis. Are there evidences that prove that it's not? In other words, is it a philosophical or scientific question?

Answer

It's not clear what sort of evidence could prove or disprove this idea. And that makes it philosophical, not scientific.

If someone were to develop an experiment by which we could distinguish between the two ideas, then the situation would change.

thermodynamics - Degrees of freedom and temperature

I quote the following lines directly from the Wikipedia page titled "Heat capacity":

"...rotational kinetic energy of gas molecules stores heat energy in a way that increases heat capacity, since this energy does not contribute to temperature."

Why doesn't rotational energy contribute to temperature?

Answer

For example, rotational kinetic energy of gas molecules stores heat energy in a way that increases heat capacity, since this energy does not contribute to temperature.

This description is misguiding in two ways.

First, the statement that

rotational energy does not contribute to temperature

makes an impression that temperature is a quantity that is closely connected with the translational kinetic energy, but not rotational kinetic energy. But that is false; according to classical theory (applicable when temperatures are high) in thermodynamic equilibrium, all quadratic degrees of freedom, translational and rotational, correspond to kinetic energy $k_BT/2$ on average.

It is only true that rotational energy does not contribute to translational kinetic energy of molecules, since the two energies are exclusive contributions to total kinetic energy.

Second, heat capacity when molecules are allowed to rotate is not higher because rotational energy does not contribute to translational kinetic energy of molecules.

It is higher because for the same temperature, such system has higher energy than system without rotation. This is because there are additional degrees of freedom, to which corresponds additional average kinetic energy.

Equilibrium implies temperature implies average energies of molecules. Value of average kinetic energies of molecules neither implies temperature exists nor implies temperature is only connected to translational kinetic energy.

mathematics - Unique Licence Plates

Note: This is a Mathematical puzzle, not a Mathematical problem. Although the answer could be solved with mathematics, lots of problems on Puzzling SE could be solved with mathematics, and the problem itself is not that mathematical. The math-puzzle tags are simply because, as I said, the puzzle could be solved mathematically.

Imagine you have 500 cars lined up in a parking lot that a buyer would like. You are the license plate creator who has to create custom plates for the buyer's 500 cars. The buyer makes a strange request. He wants you to create 4-digit numbers on each license plate in such a way that the first three digits and the last three digits cannot be used again for any other cars. (If somebody could improve that explanation, go for it.)

Here's an example: Say you make this license plate:

2465

The license plate's first three digits are 246 and the last three digits are 465. That means that when you are creating the other 499 license plates, these numbers, 246 and 465 cannot be used . Say you were making another license plate:

8246

The 824 wold be fine, but the buyer would not accept this license plate because the 246 was used previously in another car's license plate.

Info: There are 1,000 different three-digit numbers that could go in the first slot and the second slot of the license plate. This means that if each car needs two numbers, then there would need to be 500 cars, which is how many that is in this puzzle.

Diclaimer: I do not know the answer to this puzzle, but was curious what the answer is, and how you would explain it. I thought this up while calculating the odds that a certain three-digit number would appear in a four-digit number section on a license plate. I am asking for answer which explains why or why not meeting the buyer's needs are possible.

If you need any help understanding, ask in the comments.

Answer

Another answer: start the license plate making process by making all possible 500 three-digit plates, where the three digits have even sum. Now to each add a last digit as so: If the first digit on the plate is $n$, use $n+1$ as the last digit (and if the first digit is $9$, use a $0$ as the last digit).

The first three digits of any plate can't be the same as any other first three digits by design. The first three of any plate can't be the same as any last three, because the first three always has even sum and the last three always has odd sum. Finally, the last three of any plate can't overlap with any other plate, since that would imply the first three would be the same (i.e. if two plates end in $340$, both plates must be $9340$).

mathematical physics - Reason for the discreteness arising in quantum mechanics?

What is the most essential reason that actually leads to the quantization. I am reading the book on quantum mechanics by Griffiths. The quanta in the infinite potential well for e.g. arise due to the boundary conditions, and the quanta in harmonic oscillator arise due to the commutation relations of the ladder operators, which give energy eigenvalues differing by a multiple of $\hbar$. But what actually is the reason for the discreteness in quantum theory? Which postulate is responsible for that. I tried going backwards, but for me it somehow seems to come magically out of the mathematics.

Answer

If I'm only allowed to use one single word to give an oversimplified intuitive reason for the discreteness in quantum mechanics, I would choose the word 'compactness'. Examples:

1. 
   

   The finite number of states in a compact region of phase space. See e.g. this & this Phys.SE posts.

   
   


2. 
   
   

   The discrete spectrum for Lie algebra generators of a compact Lie group, e.g. angular momentum operators. See also this Phys.SE post.

   
   


3. 
   

   On the other hand, the position space $\mathbb{R}^3$ in elementary non-relativistic quantum mechanics is not compact, in agreement that we in principle can find the point particle in any continuous position $\vec{r}\in\mathbb{R}^3$. See also this Phys.SE post.