I would be most thankful if you could help me clarify the setting of primary constraints for constrained Hamiltonian systems. I am reading "Classical and quantum dynamics of constrained Hamiltonian systems" by H. Rothe and K. Rothe, and "Quantization of gauge system" by Henneaux and Teitelboim. Consider a system with Lagrangian $L(q,\dot{q})$ and define the momenta $p_j = \partial L/\partial{\dot{q}_j}$, with $j$ from 1 to $n$ (for $n$ degrees of freedom), and the Hessian $W_{ij}(q,\dot{q}) = \partial^2 L/\partial{\dot{q}_i}\partial{\dot{q}_j}$. Let $R_W$ be the rank of the Hessian $W_{ij}$. Assume that $L$ is singular and $R_W < n$. In order to ask the question, I'm now quoting how the Rothes describe the setting of primary constraints on pages 26 and 27 of their book. > Let $W_{ab}$, ($a,b = 1,...,R_W$) be the largest invertible submatrix of > $W_{ij}$, where a suitable rearrangement of the components has been carried > out. We can then solve the eqs. $p_j = \partial > L(q,\dot{q})/\partial{\dot{q}_j}$ (1) for $R_W$ velocities $\dot{q}_a$ in > terms of the coordinates $q_i$, the momenta $\\{p_a\\}$ and the remaining > velocities $\\{\dot{q}_\alpha\\}$: $\dot{q}_a = > f_a(q,\\{p_b\\},\\{\dot{q}_\beta\\})$, with $a,b = 1,...,R_W$ and $\beta = > R_W + 1,...,n$. > > Inserting this expression into the definition of $p_j$, one arrives at a > relation of the form $p_j = h_j(q,\\{p_a\\},\\{\dot{q}_\alpha\\})$. For $j = > a$ ($a = 1,...,R_W$) this relation must reduce to an identity. The remaining > equations read $p_\alpha = h_\alpha (q,\\{p_a\\},\\{\dot{q}_\beta\\})$. But > the rhs cannot depend on the velocities $\dot{q}_\beta$, since otherwise we > could express more velocities from the set $\\{\dot{q}_\alpha\\}$ in terms > of the coordinates, the momenta, and the remaining velocities. This is where the presentation from Rothe stops, and my concern is that equations of the form $p_\alpha = h_\alpha (q, \\{p_a\\},\\{\dot{q}_\beta\\})$ (2) with all $\\{\dot{q}_\beta\\}$ present can still be possible, and yet one cannot solve for more velocities from the set $\\{\dot{q}_\alpha\\}$ in terms of the coordinates, the momenta and the remaining velocities if the conditions stipulated in the Implicit Function Theorem are not met, for not all equations of the type (2) can be solved implicitly for $\\{\dot{q}_\alpha\\}$. Therefore it is not proven that there are $(n - R_W)$ primary constraints of the form $\phi_\alpha (q,p) = 0$. Henneaux and Teteilboim even state that these $(n - R_W)$ constraints of the form $\phi_\alpha (q,p) = 0$ are functionally independent, but give no justification to this statement. I would be most thankful if you could help clarify my above concern and also if you could clarify the statement by Henneaux and Teitelboin as to the fact that the constraints are functionally independent. Thank you!


Is General Relativity applicable for all coordinate systems?

My understanding was that relativistic physics can be expressed in any _inertial_ coordinate system, but not arbitrary systems. That is, no experiment can determine if we are "still" or "moving" at a constant velocity; but we can determine if we are accelerating, or moving in a circle (which by definition involves constant acceleration perpendicular to the current velocity). Thus, we can clearly state that the Earth is _orbiting_ , and can't view it as relativisticly stationary. But, to my shock, I recently came across this text http://books.google.com/books?id=lWEmNBaHCJMC&pg=PA211&dq=einstein+infeld+physics+ptolemy+copernicus&hl=en&ei=dWZ_TubbKqn20gH8hNjSDw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCwQ6AEwAA#v=onepage&q&f=false (which has Einstein as a coauthor) which, on page 212, seems to say that although special relativity requires an inertial coordinate system, general relativity does not! And that therefore we can state that the Earth is stationary and the Sun orbits it! I would reject this as pseudoscientific bunk, if not for the authors of the book.


Can a planet have a larger diameter than Jupiter?

I have heard it said that if mass were added to Jupiter, then only its mass and density would increase but the diameter would stay about the same. Is this the case for Jupiter or some property of Jupiter specifically, or is 150,000 km pretty much the largest diameter that a non-fusing object can achieve? I am specifically interested in the case of a physically larger object (planet) orbiting a smaller but more massive object (star).


How do I use the electro-optic tensor?

I would like to calculate the performance of an optical phase modulator, in which a varying electric field across a crystal modulates the effective refractive index of light passing through the crystal. The effect depends on the crystal orientation, polarization (i.e. direction of the E-field of the optical beam), and the direction of the applied electric field. The dependence on these vectors is encoded into the electro-optic coefficients, which form the electro-optic tensor. My question: How do I apply these coefficients to give the magnitude of the electro-optical effect (in radians of optical phase per applied volt of potential difference) when the light is polarized in direction $i$, propagating in direction $j$, with an applied electric field in direction $k$? In short, how do I _use_ the electro-optical coefficients? For concreteness, I am interested in lithium niobate. A datasheet gives the coefficients as: ![enter image description here](http://i.stack.imgur.com/2Tynp.png)


Can someone explain to me inertia?

I'm a little new to physics and I'm not quite getting the concept of inertia. Can someone help me?


Normal modes of oscillation: how to find them

Are normal modes the eigenvectors of the matrix $(\omega ^2 T- V)$ where $T$ is the matrix of kinetic energy and $V$ is the matrix of potential energy? Is it the only way to express them? How can I express them using the coordinates that I have choosen at the beginning of the exercise?


Experimental evidence for color-neutral gluons $( (r\bar{r}−b\bar{b})$ and $(r\bar{r}+b\bar{b}−2g\bar{g}) )$

Is there any experiment/measurement that would have a different outcome if one of the following scenarios is applied:   1. The two color-neutral** gluons would not exist   2. Those gluons would have very large masses ** we are talking about the 2 gluons not exchanging color, conventionally written as $(r\bar{r}−b\bar{b})/\sqrt2$ $(r\bar{r}+b\bar{b}−2g\bar{g})/\sqrt6$ EDIT: Since I am now aware that 'color-neutral' does not exist, the question only makes sense if stated as 'What would be the experimental evidence if color-symmetry was broken in some way? (such that 2 of the gluons were singled out)'


Do I have the meaning of the property temperature correct?

OKay my book just starts out talking about the vague definition we have for temperature and we ended up with the Zeroth law of Thermodynamics which states: > Two systems are in thermal equilibrium if and only if they have the same > temperature. So does that mean, the (brief) meaning of the property _temperature_ is **a measure of whether one system is in thermal equilibrium with another**?


Photonics: Slab As a Lens

The question can be found here: http://gyazo.com/fc4d26cd35e6ce368ad2a8ed504f1dcc The refractive index it references can be found here: http://gyazo.com/94fd2f3b5ea7da9226c3acd56b0024c1 I'm not really sure where to get started on the "showing" part of the question. I anticipate that Snell's Law will be used somehow. I'm not looking for a full answer, just a push in the right direction.


Thought experiment regarding an object approaching a mirror

Here's a thought experiment I came up with in class today when my mind drifted (I however highly doubt I'm the first to think about this since it is pretty rudimentary) : **Let's say superman approaches a (flat) mirror at 200,000 km/s. At what speed does superman's image approach him?** This has given me a big headache. I have pondered a lot of solutions, but none of them really convinced me. What is the correct solution to this thought experiment?


Are there any modern textbooks on statistical mechanics which don't ignore Gibbs' analysis of the microcanonical ensemble?

I have lately been reading Gibbs' book _Elementary Principles in Statistical Mechanics_ , and I'm surprised how much in that book seems to have been ignored by later textbook writers. In particular, Gibbs looks carefully into the question of whether in the microcanonical ensemble we can sensibly define entropy and temperature, and his answer, to paraphrase, is "meh, kind of". Gibbs pays close attention to exactly deriving properties of systems with any number of degrees of freedom (not just macroscopic systems), and shows that the canonical and grand canonical ensembles give perfect explanations for thermodynamic equations, _even extending them to small systems_. This is not so for the microcanonical ensemble:   * There are two valid definitions of microcanonical entropy (phase space "surface" entropy (Shannon/Boltzmann entropy) and phase space "volume" entropy, to use the modern terminology). These are both valid since they become equivalent in the thermodynamic limit.   * Neither definition, when differentiated against energy, leads to a satisfactory value of temperature, because two microcanonical systems, when they are combined and allowed to exchange energy, do not maintain the same temperature that they had before. (Except in the thermodynamic limit.)   * For a system with 1 or 2 quadratic degrees of freedom (e.g. particle in a 1D or 2D box), these temperatures behave super strangely. As in, being always negative or always infinite, and the system becoming colder with increasing energy. Again, the thermodynamic limit escapes these problems. It seems like the only value of the microcanonical ensemble is in the thermodynamic limit where these two expressions of entropy and temperature become equivalent, and start to behave properly. On the other hand, in a lot of textbooks the microcanonical ensemble is taken to be somehow the "fundamental" ensemble, even though the ensembles become equivalent in the thermodynamic limit anyway! So, putting the question in a slightly different way, are there any modern textbooks on statistical mechanics which are as careful as Gibbs, or are learners simply expected to read Gibbs for the "real thing"? I would guess that proper treatment of small systems is a big deal for, e.g. nanotechnology.


Commonalities among string theories?

So string theory is a developing theory with presumably different variants. What is the commonality among all these variants? What makes the next modification of a current string theory another string theory, as opposed to a completely new theory? More specifically, is there a general mathematical structure that is common to all the theories that are considered to be string theories?


Decomposition of representations of the Virasoro algebra under $sl(2)$

The Virasoro algebra has a finite $sl(2)$ sub-algebra generated by $L_{-1}$, $L_0$ and $L_{+1}$. Let's consider a unitary highest weight representation of the Virasoro algebra with conformal weight $h>0$ and central charge $c>1$. How does this representation decompose under the $sl(2)$ sub-algebra? It is clear that there is an invariant $sl(2)$ sub-module of weight $h$ consisting of the highest weight state $|h\rangle$ and the descendants $L_{-1}|h\rangle$, $L_{-1}L_{-1}|h\rangle$, $\dots$ However, there are many other descendants, such as $L_{-2}|h\rangle$, $L_{-3}|h\rangle$, etc. What $sl(2)$ modules will these states sit in? In particular, is the full $sl(2)$ representation completely reducible, so that it can be written as a sum of irreducible $sl(2)$ representations? Can anything general be said about the multiplicity of the $sl(2)$ representations appearing in the decomposition?


Writing Dirac Mass Term For Massive Neutrinos

How does one write down the following Dirac mass term for a collection of "massive" neutrinos? \begin{equation} -[\overline{(\psi_R)}M_D\psi_L+\overline{(\psi_L})M^\dagger_D\psi_R] \end{equation} I can write down the mass term as $-\bar\psi M \psi$, where M is a diagonal matrix with mass eigenvalues, $\psi=(\psi_1,...,\psi_n)^T$ is a vector in flavor space and defining adjoint as $\psi=(\bar \psi_1,...,\bar\psi_n)$ and any $\psi_i=\begin{pmatrix} \psi^i_L\\\ \psi^i_R \end{pmatrix}$. My confusion is, with these I can show the above expression except I get only $M_D$ and not $M_D^\dagger$, but in the literature of neutrino mass people write it this way.


Anomaly cancellation and fermion number violation

In the standard model, an axial $SU(3)$ currents has anomaly which after quantization leads to the fermion number violation. However, taking all the fermions into account we note that the anomalies cancel. Does the cancellation imply that the net fermion number violation is zero?


Lorentz model and energy exchange

The Lorentz model, describing the electron of the atom as an harmonic oscillator forced by an oscillating electric field $\vec{E}$, shows that the dipole moment $\vec{D}$ obeys the following equation in the stationary regime: $$\vec{D}=\frac{q^2}{2m\omega_0}\frac{\vec{E}}{\omega_0-\omega-i\gamma_d}$$ where $\omega_0$ and $m$ are the natural frequency and mass of the electron, $\gamma_d$ is the dissipation introduced by hand to take account of the energy emitted by the electron when accelerated (and also its collisions). My trouble is that one finds that the energy transmitted by the electron to the field by unit of time, given by: $$P=-\vec{E}.\frac{d\vec{D}}{dt}$$ is always negative, implying that the electron cannot "feed" the field, however how much it is radiating. Can it be attributed to the fact that the electron energy is in first place coming from the field, and thus, our electron cannot give more than it receives? In what cases does the electron sources the background field?


Finding the acceleration of Block attached using tricky string setup

Below shown is a setup, and block B starts from rest and moves towards right with a constant acceleration. Does the acceleration differ for the blocks ? I am a bit confused because of the tricky string setup. If after time t, the velocity of A with respect to B becomes v, then what would be the acceleration of A ? My understanding is : After time t, Block B will be brought to rest instantaneously resulting in non-zero relative velocity which otherwise will continue to be 0. ![Block String Setup](http://i.stack.imgur.com/ErLPK.png)


Has BCS Cooper pair condensate been observed in experiment?

Feshbach resonance in s-wave scattering states a BCS Cooper pair condensation at B-field just above the resonance where the scattering length a <0\. Just wondering if the condensation has been observed in an experiment in any group? I was looking at a notes written in 2005 which claimed that this observation was the holy grail, and I seem not to be able to find a paper claiming that particular observation in experiment.. Many thanks :) Simon


Fresco in his "Future By Design" introduces drawing of underwater vessel and its front system of generating air bubbles. Is idea energy efficient?

I recently saw Fresco's Future By Design and noticed something miniature to investigate. My notice regards about one of his illustration he describes in the documentary. Here is the link on youtube, timecode added where it starts with stuff that cought my eye. http://www.youtube.com/watch?feature=player_detailpage&v=5zn8MRKOskw#t=2812s He introduces front system of generating air bubbles in one of his underwater vessel. It's usage would be for reducing resistance while the vessel is moving thru water. I recon he invested some time to devise this illustration and this system of air bubbles fits his high tech vision perfectly and my intention with this question is to kinda have his work evaluated xD Now my question: can this system be constructed to be energy efficient? i'am no engineer but i have an opinion that in reality, its invested/returned energy ratio has to be high and therefore ridiculous to be used.


Calculate TEC efficiency

I am currently doing a small experiment using a Peltier to cool 250ml of water. My aim is to achieve the cooling of 250ml from 23degC to 8degC in under 20minuets which I have done by strapping the Peltier to the underside of an aluminium cup and a large air cooled heat sink and fan to the hot side of the Peltier. The Peltier was consuming 4A at 6V The heatsink reached 31degC (average from top to bottom) I am now looking at the theory behind the efficiency of it. I have read that (currently) peltiers can only achieve a maximum efficiency of 15% to me that means for every watt of power in you get 0.15watts cooling power, the rest is wasted in heat and work. When I researched how they got to that figure it came up with co-efficiency of performance (COP). I took the figures from my experiment and applied the COP formula and got somewhere around 0.47COP as COP is a ratio that means I should be getting somewhere close to 50% efficiency. Can someone explain this to me please?? I would also like to rerun the experiment after tweaking some values to find its optimum performance, but I need to address the efficiency thing first.


Divergent path integral

What does it mean to have a divergent path integral in a QFT? More specifically, if $$\int e^{i S[\phi]/\hbar} D\phi (t)=\infty $$ What does this mean for the QFT of the field $\phi $? The field $\phi$ has action $$S[\phi]=\int\left(\frac{1}{2}\partial_\mu\phi\partial^\mu\phi-V(\phi)\right)\mbox{d}vol$$ where we use Minkowski signature $(+,-,-,-)$.


Energy difference between symmetric and antisymmetric wavefunctions

Is there any energy difference between a particle in a symmetric wavefunction and an identical particle in an identical potential but in a state with an anti-symmetric wavefunction? Or is it case-dependent? The full problem statement (from this assignment) is ![problem](http://i.stack.imgur.com/QAzs5.png)


Why do people say "Bosons are either gapped or condensated, except physical principle protected cases (Goldstone boson, photon)."?

Bosons are either gapped or condensated, except physical principle protected cases (Goldstone boson, photon, etc.). I read this in a paper (version1 of http://arxiv.org/abs/1404.3728v1, 1st page 1st column) and in other lecture slides (page 6 of http://discor2013.files.wordpress.com/2013/12/lee.pdf). This slide appears to talk about some mean field result of spin liquid state. This statement is not really pertinent to the mainstream of that paper. But it sounds rather intriguing and necessary to be further clarified. So could anyone make it clearer? Thanks!


How to transfer mechanical power from the inside of a vacuum chamber to the outside while maintaining a seal?

In a vacuum chamber how would one transfer mechanical power (either rotation or linear) from inside to the external environment? I'm working on an idea for a new/different type of motor that would require an evacuated internal atmosphere and am wondering how to transfer the generated motion outside the case. I have thought about using rotating magnetic coupling on either side of a thinned wall section but don't think it would scale well for larger versions of the motor (perhaps powering a bike or car). I expect with a high enough budget (which I don't have) that an extremely high-tolerance mechanical seal could be machined but am wondering if there are any solutions that could sidestep the high tolerances by thinking laterally.


Momentum in Free Scalar Field

I have a few issues with making the transition between these: $\phi(x)=\int{\frac{d^3p}{2\pi^3}\frac{1}{\sqrt{2\omega_{\vec{p}}}}(a_{\vec{p}}e^{i \vec{p} \vec{x}}+ a^{\dagger}_{\vec{p}}e^{-i \vec{p} \vec{x}}})$ $\pi(x)=\int{\frac{d^3p}{2\pi^3}(-i)\sqrt{\frac{\omega_{\vec{p}}}{2}}(a_{\vec{p}}e^{i \vec{p} \vec{x}}- a^{\dagger}_{\vec{p}}e^{-i \vec{p} \vec{x}}})$. So from the Lagrangian of the real scalar field, we get $\pi(x)=\partial_0 \phi(x)$ which just means differentiating our $\phi(x)$ w.r.t time. But where does the $-i$ come from (as in the minus sign)? Also aren't the exponentials independent of time as $\vec{p} \vec{x}$ isn't a four vector? So now I'm wondering where the $\omega_{\vec{p}}$ has come from as well!


How to calculate the partial entropy in a fully connected ising system

I'm trying to reproduce a calculation that should lead to the partial entropy in a fully connected ising model for the high-temperature range ($\beta < 2$) in the thermodynamic Limit ($N \rightarrow \infty$). Although this is a very tempting goal -- the partial entropy would lead to an analytic expression for the mutual information -- I'm still failing and after a very frustrating week my last hope is to find help this way. The model is described by the hamiltonian $H=-\frac{S^2}{N}$. Here $S:=\sum_{i=1}^N s_i$ is the sum of all spins and $N$ is the number of spins in the model. The partial entropy then is $H_A = - \sum _{p_A=0}^{N_A} \binom{N_A}{p_A} R(p_A) \log_2 R(p_A)$ with $R(p_A):= \sum_{p_B=0}^{N_B} \binom{N_B}{p_B} \frac{\exp \left[ N \beta \left( \frac{p_A}{N} + \frac{p_B}{N} - \frac{1}{2} \right)^2 \right]}{Z}$ where $N_A$ and $N_B$ are the number of spins the subsystems $A$ and $B$ respectively and $p_i$ are the number of spins of the subsytsem $i$ pointing in the $-x$ direction. What i'm failing now is to reproduce that this leads to $H_A = \tau N - \frac{1}{2 \log 2} \left\\{ \frac{\beta \tau}{2 - \beta} + \log \left[ \frac{2-\beta}{2-\beta(1-\tau)} \right] \right\\} + O(1/N), \quad N \rightarrow \infty,\ \beta < 2, \tau := \frac{N_A}{N}$. My last attempt was to apply the replica trick to get rid of the $\log_2$ in $H_A$ and then use some kind of saddle-point method for the resulting $n+1$-fold Integral to solve it in the thermodynamic limit. This looked promising on the first sight but i can't find a closed analytic expression for the determinant of the Hesse-Matrix. The analytic expression is formula #26 in "J. Wilms, J. Vidal, F. Verstraete, and S. Dusuel. Finite-temperature mutual information in a simple phase transition. _Journal of Statistical Mechanics: Theory and Experiment_ , 2012:P01023, 2012."


Application of diffraction problem!

Here is a problem that I am working on, which is the applying the concepts of diffraction to the setting of the sun: > Air has a small, usually negligible index of refraction. It is 1.0002926. > This causes the Sun to actually be below the horizon when it appears to be > just on the verge of sinking below it. > > Suppose you are on the sea-shore watching the Sun apparently sinking into > the ocean. When only its upper tip is still visible, by what fraction of the > diameter of the Sun is that tip actually already below the surface? > > As an approximation, take the earth's atmosphere as being of uniform density > out to a thickness of 8.600 km, beyond which there is no atmosphere. This > means that, with the Earth's radius being 6400. km, your line of sight due > West along the ocean surface to the horizon will intersect this "upper > surface" of the atmosphere at about 331.9 km from your eye. (The diameter of > the sun subtends 0.5000 degrees at your eye). This is how I've attempted to model the problem, but I'm not sure if it's correct... ![problem](http://i.stack.imgur.com/mOsYQ.png) I'm not really sure how to approach this problem.... Can anyone please help me?


Can I take heat from the air and convert it to electricity?

Its a summer day and the air in my house has been heated up. I could switch on my air conditioning, but then I'd be using energy from the grid in order to reduce the amount of energy in my house. What I'd much rather do is capture the heat energy from the air, so cooling it down, and turn that heat energy into electricity. (So I could store it in a battery or sell it to the grid or use some other means of disposing of it.) Is such an apperatus possible within our present understand of nature? Why or why not?   (I presume not, as people would be selling kits.) Just to pre-empt a possible answer: I've seen similar conversations in the past and "inefficient" has been given as the answer as to why not. I don't understand this as an objection, because the heat energy is effectively free. The sun rudely went and gave me an excessive amount of heat energy without being asked to. Is converting heat to electricity really so ineffiecient that I'd be better off running air conditioning?


Understanding Calculus Notation in Physics

I have just started a first-year calculus-based physics course about electromagnetism and waves. I am having trouble understanding what calculus notation means in the context of physics. Here is a paraphrased example from Fundamentals of Physics 9th ed. by Halliday et al. that considers finding the electric field at point P (refer to the figure below): Let $ds$ be the arc length of any "differential" element of the ring. $dq=\lambda~ds$ is the total charge of the ring, where $\lambda$ is linear charge density. The magnitude of the field is given by $$dE=\frac{1}{4\pi\epsilon_0}\frac{dq}{r^2}=\frac{1}{4\pi\epsilon_0}\frac{\lambda~ds}{r^2}$$ Since the x-components of the field are balanced at point P, after the application of geometry, the equation becomes $$dE\cos{\theta}=\frac{z\lambda}{4\pi\epsilon_0 (z^2+R^2)^{3/2}}ds$$ The author then puts an integral sign in front of both sides and integrates around the circumference of the ring: $$E=\int dE~\cos{\theta} = \frac{z\lambda}{4\pi\epsilon_0 (z^2+R^2)^{3/2}}\int_0^{2\pi R}ds =\frac{z\lambda(2\pi R)}{4\pi\epsilon_0 (z^2+R^2)^{3/2}}$$ In math, I was taught that Leibniz's notation ($dy/dx$) is not a fraction. After a little bit of research, I found that it sometimes _acts_ as a fraction, even though it is _not_ a fraction when defined with limits. Nevertheless, what should I think of when I see $dq$, $ds$, and $dE$? What does "differential" mean as used in the text above? So far it has made sense for me to think of $dq$ as an "infinitesimal" charge (for example), even though I was taught calculus using limits (I know very little about infinitesimals). Secondly, the way the integral is taken seems informal and confuses me. I understand that it represents the sum of the electric field caused by all the charge around the ring, but how can you just put an integral sign in front of both sides and take the integral with respect to whatever $dE$ or $ds$ that happens to be there? Also, the integral on the left side is indefinite whilst the one on the right is definite. How should I view this kind of material? ![figure](http://i.stack.imgur.com/SesBB.png) I am unsure if this question belongs on the math stack exchange or this one.


why is a resistor frequency independent

I had a doubt that why is a resistor, frequency independent? Since, as frequency increases the movement of electrons increases so heat increases which causes change in resistance. So my question is why a resistor offers same resistance to every frequency. Any Clarification on this is appreciated. Thanks.


Obtaining isotope stability

For a given isotope, one can obtain the binding energy using the semi- empirical mass formula. For example, ![From Codecogs](http://i.stack.imgur.com/U4is7.png) has a binding energy of 1782.8 MeV. From this information, how can the likelihood of the isotopes stability be found?


Why is the critical density of the Universe non-zero?

I'm missing something and I wondered if someone could enlighten me. As I understand it, Einstein's equations say, among other things, that matter curves spacetime. I also understand, the Universe has a 'critical density', defined to be the mass (or energy) density required for space to have zero curvature (found by solving Friedmann's equations). Why does the Universe need a non-zero density (the critical density) for zero curvature? Many thanks.


What is the meaning of following expresion $C=\frac{\delta Q}{dT}$ mathematicly

Our professor raised the following question during our lecture in Statistical Physics (even so it's related to Thermodynamics): Many text books (even wikipedia) writes wrong expressions (from mathematical point of view) for Heat Capacity Coefficient, and the right way to wright it is as following: $$C=\frac{\delta Q}{dT}$$ But as we see it is neither usual differential, nor a functional derivative, so the question what is this? I couldn't find the answer in math books, and it is true that many text books writes it in very different ways mixing exact and inexact differentials, so anybody have a clue what is the right expression for c and why from mathematical point of view?


Curvilinear coordinate system around body of revolution

In _Boundary-Layer Theory_ by Schlichting he gives the boundary-layer equations for a body of revolution according to the paper by Boltze$^1$. Unfortunately, this paper is in German. He apparently uses the following local curvilinear coordinate system to derive the boundary-layer equations: ![Fig. 11.6 from *Boundary-Layer Theory*](http://i.stack.imgur.com/zPgH8.jpg) I'm not specifically interested in deriving the boundary-layer equations for a body of revolution, however, I am working on a problem with a very similar geometry and am having trouble wrapping my head around the coordinate system. **Question:** For the locally orthogonal basis vectors $\mathbf{e}_x$, $\mathbf{e}_y$, and $\mathbf{e}_\theta$, what are the _gradient_ , _divergence_ , and _Laplacian_ operators? This is what I have tried: $$x = \hat{x}$$ $$y = \hat{y}$$ $$r(x)\cos\theta = \hat{z}$$ where $\hat{x}$, $\hat{y}$, and $\hat{z}$ are coordinates in a Cartesian system with the same origin. Lamé's coefficients are then: $$h_x = \left\vert\left(\frac{\partial \hat{x}}{\partial x}, \frac{\partial \hat{y}}{\partial x}, \frac{\partial \hat{z}}{\partial x}\right)\right\vert = 1$$ $$h_y = \left\vert\left(\frac{\partial \hat{x}}{\partial y}, \frac{\partial \hat{y}}{\partial y}, \frac{\partial \hat{z}}{\partial y}\right)\right\vert = 1$$ $$h_\theta = \left\vert\left(\frac{\partial \hat{x}}{\partial \theta}, \frac{\partial \hat{y}}{\partial \theta}, \frac{\partial \hat{z}}{\partial \theta}\right)\right\vert = \left\vert r(x)\sin\theta\right\vert$$ With these coefficients it is straightforward to define the operators since this system is orthogonal (for example here are the formulas). I do not think that the results above are correct though since it compares to a local Cartesian system at each point. What are the correct transformations?   1. Boltze, Ernst. Grenzschichten an Rotationskorpern in Flussigkeiten mit kleiner Reibung. Georg-August-Universitat zu Gottingen., 1908. **UPDATE:** For any future Googlers, the correct transformation is: (based on Fig. 4 of @Floris's answer, $\xi$ is aligned with $U_\infty$ in the picture above) $$\xi = x\cos\phi - y\sin\phi$$ $$\eta = -(r(\xi) + y\cos\phi + x\sin\phi)\sin\theta$$ $$\zeta = (r(\xi) + y\cos\phi + x\sin\phi)\cos\theta$$ where $\phi$ is the local angle of the body with respect to the axis of rotation, i.e. $\phi = arctan(\frac{dr}{dx})$. The paper by Boltze assumes this angle is small. The Jacobian is then: $$J = r(\xi) + y\cos\phi + x\sin\phi$$ which is equivalent to Boltze's value for small $\phi$ and $y \ll r(\xi)$. And the Lamé coefficients are: $$h_x = 1$$ $$h_y = 1$$ $$h_\theta = r(\xi) + y\cos\phi + x\sin\phi$$


Relationship between lightlike and spatial compactification

The compactification of a spatial dimension, say $x^1$ given by the identification $x \sim x^1 + 2\pi R$ is said to be related to the lightlike compactification by a Lorentz boost : $$ \left( \begin{array}{c} x^0 \\\ x^1 \\\ \end{array} \right) \sim \left( \begin{array}{c} x^0 \\\ x^1 \\\ \end{array} \right) +2\pi \left( \begin{array}{c} -R \\\ R \\\ \end{array} \right) $$ What exactly is the relationship between them? How can I see this? And what does it mean from a physics point of view?


Heisenberg picture saves locality in QM?

David Deutsch recently released a paper titled "The vindication of locality" where he argues that by using the Heisenberg picture rather than the Schroedinger picture, you can "save" locality. The paper is here: http://arxiv.org/abs/1109.6223 Is Deutsch wrong or has he indeed "proven" locality?


Decomposition of this wave function in eigenfunctions

I have this wave function of a system on a central potential: $V(r)$: $$\Phi(x,y,z)=C(x+y+z)e^{-\alpha r^2}.$$ And I'm asked a few things about probabilities. I don't have problems with that, because I understand how it's done. The main part is decomposing as a sum of eigenfunctions of the form. Changing the variables $x,y,z$ by their spherical forms, we have: $$\Phi = Cr(\sin\theta\cos\phi+\sin\theta\sin\phi+\cos\theta)e^{-\alpha r^2}.$$ The part that depends on $\theta,\phi$ can be expressed as: $$\sin\theta\cos\phi+\sin\theta\sin\phi+\cos\theta=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell} a_{\ell m}Y_\ell^m(\theta,\phi).$$ Being $a_{\ell m}$ the scalar product of our function with the eigenfunctions. What I don't understand is that when we did it in class the teacher said that it was necessary to do it only for $\ell=0,1$, and that we didn't need to go any further. I don't remember why this was possible to say before making any calculations. Why is it?


How many kinds of topological degeneracy are there?

Here I want to summarize the various kinds of topological ground-state degeneracy in condensed matter physics and want to know whether there exists any other kind of topological degeneracy. For convenience, let's consider a 2D lattice system with $N$ lattice sites, and we may consider the finite system with open boundary condition(OBC) or periodic boundary condition(PBC). Now there are two kinds of topological degeneracy: (1)The topological degeneracy is **approximate** for finite $N$ no matter with OBC or PBC, and it becomes **exact degenerate** only under the thermodynamic limit($N\rightarrow \infty $). **Example:** FQHE. (2)The topological degeneracy(>1) is **exact** for **any finite $N$** with **PBC(on a torus)** , and it's **nondegenerate** for **any finite** $N$ with **OBC**. **Example:** Kitaev's toric code model. Is there any other kind(in the above sense) of topological degeneracy?


Free path distribution

I'm studying statistical mechanics, and I'm trying to resolve some problem known from my thermodynamics course. So I want to _calculate mean free path_ for particles with a concentration $n$ and scattering cross section $\sigma$ using **only statistical approach**. I understand that for the first step I need to find the probability distribution of the free path of the molecules of gas between successive collisions and then calculate the average value. But here I have some troubles, because I have never solved something like this. I would be grateful if someone would give me some links or give some ideas.


Can we show that time is orthogonal to space?

It's easy to show that the time we measure is "in a different direction" from the space directions we measure. However, it's not immediately obvious to me that these directions are **orthogonal**. How do we prove that any directions are orthogonal? What I came up with is the following. [For short distances/does this cause trouble?], we can use the pythagorean theorem. If I have two directions in space, I can simply travel 1 meter in each of the two directions, and then measure the distance between the two endpoints. The result is sqrt(2) meters if and only if your two directions were perpendicular. ![Finding whether two directions are orthogonal](http://i.stack.imgur.com/I4rIu.png) Easy! My question is: **can this method be used to prove that our concept of time is orthogonal to our concept of space?** I have an idea of where to start. We would need to move in space by 1 meter, while moving in time by 1 meter (converting time into meters using c). Then we would need to measure whether the distance traveled was sqrt(2) meters. I'm not sure I know what that would mean. Hopefully this question makes sense; thanks for your assistance!


What happens to the physical properties of electrons after diffraction?

Particle Wave duality shows us that waves and particles are the same thing. Therefore electrons can be viewed as both particles and waves. The wave properties of electrons can be seen in the double split experiment where a diffracted electron interacts with itself. If a single electron was diffracted, what would happen to it's physical properties? How could the mass spread out like the wave? Similarly how would the charge and spin spread out? These properties are usually considered to be contained in particles, yet if a single electron was diffracted, it surely could not become many particles spreading out. Also, what would happen if something interacted with this 'electron wave' causing part of it to act like a particle again? The wave can't be able to turn back into a particle after spreading out so much, yet it must still be able to act like a particle for certain interactions.


Rotational inertia of a sphere and a cube?

Can someone explain which would have more rotational inertia, a sphere or a cube, and why? Suppose they have the same mass, the side of the cube is equal to the diameter of the sphere, and the cube's rotation axis is perpendicular to two of its faces. Which one would have more rotational inertia about an axis through the center of mass?


i can't detect what wrong have i done in a kinematics problem

a question was given to us in assignment which was as follows > A person moves 20 metres towards north then 30 metres towards east and then > moves 30 x 1.414 metres towards south west. The whole trip takes 5 seconds. > What is the person's average velocity for this whole journey? i just squared every magnitude of velocity in all the three directions and then I took square root of that sum doing so what i got is this number `55.735267112` so this will be the total displacement of that person and dividing this with the given time [5in this case] we get the answer as 11.1 but the answer is 2 m/s can any one suggest me where is my concept going wrong


What does "soft" in "soft symmetry breaking" mean?

For example it is stated that if supersymmetry breaking is _soft_ then stability of gauge hierarchy can be still maintained.


Born rule and unitary evolution

Is the Born rule a fundamental postulate of quantum mechanics, or can it be inferred from unitary evolution?


Why are these specific stress invariants chosen?

I've seen these invariants of the Cauchy stress tensor $S$ defined in multiple places: $$J_1 = \lambda_1+\lambda_2+\lambda_3 = Tr(S)$$ $$J_2 = \lambda_1^2+\lambda_2^2+\lambda_3^2$$ $$J_3 = \lambda_1\lambda_2\lambda_3 = \det{S}$$ where $\lambda$ are the eigenvalues of the stress tensor. But I suppose we could use almost any three functions of the eigenvalues. Is there any specific reason to choose these, other than the fact that they can be calculated without first finding the eigenvalues?


How do photons 'connect' during wireless connection

So wireless router broadcasts a signal and then your device searches. So what actually happens when the photons 'meet' it's kind of like saying, 'ah your one of us, so we will follow you, show us the way' It's so bizarre, how do photons connect during wireless connection?


In coordinate-free relativity, how do we define a vector?

Relativity can be developed without coordinates: Laurent 1994 (SR), Winitzski 2007 (GR). I would normally define a vector by its transformation properties: it's something whose components change according to a Lorentz transformation when we do a boost from one frame of reference to another. But in a coordinate-free approach, we don't talk about components, and vectors are thought of as immutable. For example, Laurent describes an observer using a timelike unit vector $U$, and then for any other vector $v$, he defines $t$ and $r$ uniquely by $v=tU+r$, where $r$ is orthogonal to $U$. The $(t,r)$ pair is what we would normally think of as the coordinate representation of $v$. In these approaches, how do you define a vector, and how do you differentiate it from things like scalars, pseudovectors, rank-2 tensors, or random objects taken from something that has the structure of a vector space but that in coordinate-dependent descriptions would clearly not transform according to the Lorentz transformation? It seems vacuous to say that a vector is something that lives in the tangent space, since what we mean by that is that it lives in a vector space isomorphic to the tangent space, and any vector space of the same dimension is isomorphic to it. [EDIT] I'm not asking for a definition of a tangent vector. I'm asking what criterion you can use to decide whether a certain object can be described as a tangent vector. For example, how do we know in this coordinate-free context that the four-momentum can be described as a vector, but the magnetic field can't? My normal answer would have been that the magnetic field doesn't transform like a vector, it transforms like a piece of a tensor. But if we can't appeal to that definition, how do we know that the magnetic field doesn't live in the tangent vector space? Bertel Laurent, Introduction to spacetime: a first course on relativity Sergei Winitzki, Topics in general relativity, https://sites.google.com/site/winitzki/index/topics-in-general-relativity


About partially polarized light and the degree of polariztion

When I was taking Optics course, I found there were several questions about polarization of light. I use the textbook of Hecht.   1. It seems that the definition of degree of polarization may be not so well-defined if $V=\frac{I_{max}-I_{min}}{I_{max}+I_{min}}$. For a elliptical polarized light, there is no natural polarized part, but still $V\neq 0$.   2. I found it hard to deal with partially polarized light. First the definition. What is the definition of partially polarized light? Light with $0<V<1$?   3. Is all partially polarized light can be presented by the superposition of a plane polarized light and a natural light? I suppose it is true but why? Is there any formal explanation?   4. Consider a real problem. Suppose there is a beam of light incident on an air-glass interface with $n_{ti}=1.5$ at a certain degree, say $30°$. Then how to characterize the reflected light or the transmitted light since they are all partially polarized. What is $V$ for these lights? I gauss the answer may be $V=\frac{|r_p|^2-|r_s|^2}{|r_p|^2+|r_s|^2}$. But I can't convince my self why this correspondes to the definition above. (This is essentially a problem in Hecht.) Thanks a lot! I am really confused with that.


Can a charged particle pass through a superconductor? I think it is impossible, because the superconductor will induce an infinite electric field which keeps the particle away.


Proof involving the fine-structure Hamiltonian of the Hydrogen atom

Given the perturbed Hamiltonian of the Hydrogen atom: $$ H=H_0+H_f=\frac{\bar{p}^2}{2m}+V(r)-\frac{\bar{p}^4}{8m^3c^2}+\frac{1}{2m^2c^2}\frac{1}{r}\frac{dV(r)}{dr}\bar{L}\cdot\bar{S}+\frac{1}{8m^2c^2}\nabla^2V(r), $$ where the last three terms are the relativistic perturbation, the spin- orbit coupling and the Darwin term and they make up the fine structure Hamiltonian $H_f$; assuming we are in the case $n=2$, I'm asked to prove that $H_f$ doesn't link the eigenvectors $| n=2,l=0,m_l=0,m_s=\pm1/2\rangle$ with the eigenvectors $| n=2,l=1,m_l=-1,0,+1,m_s=\pm1/2\rangle$. The only thing that comes to mind is to apply $H_f$ to each of the first two vectors and check that the output kets are only linear combinations of those two. I applied the separate parts of $H_f$ to those two vectors and found immediately that the relativistic part (the one with $\bar{p}^4$) doesn't mix the subspaces. But I'm stuck with the spin-orbit term: my guess is that I should use the Clebsch-Gordan coefficients to change to the $|n,l,j,m\rangle$ basis, but I'm not sure how to do this since I've never calculated CG coefficients. My questions are:   * Am I on the right track? If so, could somebody point out how to calculate the CG coefficients? If I'm not on the right track, what could I do?   * Also, the Darwin Hamiltonian ends up being proportional to $\delta(\bar{r})$. Is the proof that it doesn't link the aforementioned kets, as trivial as it looks? It just multiplies the kets and is thus its own eigenvalue? PD: I forgot to mention earlier that potential is Coulombian: $V(r)=-e^2/r$, which gives the delta in the Darwin term.


Relating the power to weather variables

Could you please help me in this problem? I have 3 independent variables, (T,H,t) , as inputs and one output P ( I have all data for these inputs and the output, done experimentally measured every hour during one year). I want to find a formula of this form: P=f(T,H,t) where t is the time in hours and it is always in the x -axis (index), T is temperature, H is humidity, and P is power. I have all the data, and when I draw them in the same graph during one year, meaning that P , T , and H vs. hours. I found that the behavior of P is oscillating, making a sinusoidal shape over the entire year: ![](http://www10.0zz0.com/2010/12/27/18/159867792.jpg) So, if I make a zoom view to this figure, for example from the 2000 th hour of the year to the 3000 th hour, it is clear that it almost has the same shape but it is oscillating. So, it keeps oscillating and increasing up to the peak point and then it starts decreasing till the end of the year. So, how can I predict the structure of the formula that relates P with T , H , and t ? What is the effect of T,H and t on P . Is there any approach that you advise me to follow? Sorry for this long question and any help from you is highly appreciated. I read many papers but I could not know how to solve the problem.


Electron indistintinguishability and handedness

I just learned that right ad left handed electrons behave in a remarkably different way under the weak interaction. Up till now I have been told that all the electrons are exact copies of one another and all this fermi-dirac statistics story. Nonetheless, since right handed and left handed electrons DO have fundamental differences, shouldn't this remove their indistinguishability?


What coefficients may be used for evaluation the heat keeping?

The final goal is to compare two objects (similar to boxes) from the standpoint of keeping the heat inside? May be there is numerical coefficient or any approaches?


How thin a filter to filter particles out of salt water to make salt for human consumption?

I have gathered some salt in the Death Valley and for the fun of it I would like to clean it and make it edible. The process I plan to follow is simple:   1. Dissolve the salt in water.   2. Boil water for 30 min to kill bacteria.   3. Filter water to remove foreign particles.   4. Evaporate water to get clean salt. At step 3, I'm planning to buy paper filters with pores of 3 µm to filter fine particles. Is it sufficient for the purpose of human consumption? I'm mostly wondering if there is a key pore size threshold beyond which it is significantly less useful to filter, or if it's just a matter of degree, the smaller the better at an consistent size/benefit ratio.


What is a gauge transformation of the metric in GR?

I'm confused by this question: Why can't General Relativity be written in terms of physical variables? I can't quite see how you can make any change to the metric without either: (a) changing the physics; or (b) changing which point in your manifold represents which physical point in spacetime. Either of these would, it seems to me, disqualify the change as being a gauge transformation, at least in the usual sense. So is there some cunning way of changing the metric that avoids either (a) or (b), or is there some sense in which (b) can still be thought of as a gauge transformation?


How did Enrico Fermi calculate the classical Fermi Problem?

From Wikipedia: Fermi was known for his ability to make good approximate calculations with little or no actual data, hence the name. One example is his estimate of the strength of the atomic bomb detonated at the Trinity test, based on the distance travelled by pieces of paper dropped from his hand during the blast. Fermi's estimate of 10 kilotons of TNT was remarkably close to the now-accepted value of around 20 kilotons, a difference of less than one order of magnitude. I have not been able to find any references explaining how he made this calculation. Please provide a reference or an example calculation. See also: http://en.wikipedia.org/wiki/Fermi_problem


When a variation of a tensor is not a tensor?

In a comment about variation of metric tensor it was shown that $$\delta g_{\mu\nu}=-g_{\mu\rho}g_{\nu\,\sigma}\delta g^{\rho\,\sigma}$$ which is contrary to the usual rule of lowering indeces of a tensor. That means $\delta g^{\mu\nu}$ is not a tensor, while $g^{\mu\nu}$ is a tensor quantity, so taking a variation leads to loss of tensor nature here. How often does that happen to other tensor quantities? Are there general rules or guidelines to know if their variations are tensors or not? **Update:** Misunderstanding clarified. I should not interpret $\delta T^{\mu\nu}$ as $(\delta T)^{\mu\nu}$ and try to apply tensor rules to indices immediately. Instead, I should take it as $\delta(T^{\mu\nu})$, and if I wish to lower indices, I should write $\delta(g_{\mu\rho}T^{\rho\nu})$ and then apply Leibniz rule to the variation as $\delta(AB)=(\delta A)B+A(\delta B)$. That would generally add some terms in case $\delta g^{\mu\nu}\ne 0$.


Error in Sear's and Zemansky's University Physics with Modern Physics 13th Edition (Young and Freeman)?

I was reading up on the Ideal Gas Equation in University Physics with Modern Physics by Young and Freeman when I chanced upon a seemingly illogical mathematical equation. Can anyone rectify this error? Or is it misunderstanding on my part? Here is the portion (Page 600, Chapter 18, Equation 18.12): > $$pV = \frac{1}{2}Nm(v^2)_{av} = > \frac{1}{3}N\biggl[\frac{1}{2}m(v^2)_{av}\biggr]$$ It should be clear that $\frac{1}{3} \neq (\frac{1}{3} \times \frac{1}{2})$.


How was Be-8's Half-Life of 7E-17 Second Determined?

Radionuclides occur with half-lives in a vast range of over 37 magnitudes as listed in this site. In question 7584, Lubos Motl explained how Gyr half-lives were determined. This method doesn't appear applicable to evaluate sub- microsecond half-lives. How are extremely short half-lives determined?


Stokes' theorem in GR

I read this formula in Sean Carroll's book of GR: $$\int_{\Sigma}\nabla_{\mu}V^{\mu}\sqrt{g}d^nx~=~\int_{\partial\Sigma}n_{\mu}V^{\mu}\sqrt{\gamma}d^{n-1}x$$ where n is the 4-vector orthogonal to the hypersurface $\partial\Sigma$ and $\gamma$ the induced metric defined by $$\gamma_{ij} = \partial_i X^{\mu}\partial_jX^{\nu}g_{\mu\nu}.$$ The only trouble I have is with the appearance of this induced metric, because I have the formula $$\int_{\Sigma}\nabla_{\mu}V^{\mu}\sqrt{g}d^nx=\int_{\Sigma}\partial_{\mu}(V^{\mu}\sqrt{g})d^nx$$ which then should give me straightforwardly $$\int_{\partial\Sigma}n_{\mu}V^{\mu}\sqrt{g}d^{n-1}x$$ by the classical Stokes' theorem. I was thinking that it may be something with the differential element: going from $d^nx$ to $d^{n-1}x$ forced the apparition of this indeuced metric but I didn't manage to figure out how so far.


Amount of material required for a pressure tank

I read the answer for the question Why is a hot air balloon “stiff”? and thought something sounded ridiculous. My engineering requirement is that the walls be strong enough. Here $T$ will be the tension (for a surface, not sure about those units) and $R$ is radius of curvature of the wall. Requirement is: $$\Delta p < 2 \frac{T}{R}$$ Let $d$ be the thickness of the wall and $\sigma$ the material tensile strength. $$T \propto d \sigma$$ This would indicate that **thickness increases linearly with scale**. That sounds ridiculous. _Why it sounds so silly_   * Volume scales as $R^3$ and surface area as $R^2$. SA x (thickness) = material volume = constant, so that implies there are no economies of scale for pressure tanks in terms of pressurized volume divided by structural materials. That sounds nonsensical. That means a chemical plant wouldn't save any materials by buying a large tank as opposed to 1,000 tiny tanks.   * Say that I have a tank shape in mind. If I build a small tank and a large tank, they will be geometrically congruent. That is, if thickness if 5% of the diameter of the small one, it will be 5% of the diameter of the large one. Please prove me wrong. And if you can't prove me wrong, please establish a physical intuition as to why this should be the case.


PN Junction. Germanium VS Silicon

As far as I understood, when we connect P and N materials, electrons from N drift to P, thus create negative ions in P and positive ions in N. => electric field (barrier potential) must depend on an amount of electrons recombined with holes in P region. => The more electrons recombined, the more ions created, the more barrier potential. BUT! Silicon has 14 electrons and 4 on the 3d level. Germanium has 32 electrons and 4 on the 4th level. => at 25C degrees, electrons in N region in Germanium must have more energy than electrons in Silicon, because less energy required to remove electron from an atom => more electrons can drift to P region, thus barrier potential for Germanium should be higher, however Vbp Germanium = 0.3 V Vbp Silicon = 0.7 V why? P.S. I am talking about P doped and N doped silicon compared to the junction between P doped and N doped germanium


Thermodynamics - convection current

please could someone help explain the mechanism of a convection current as I keep finding conflicting explanations. Is a convection current caused by differences in density or differences in pressure? Most seem to say it is caused by differences in density (more specifically buoyancy) since you cannot get a convection current in zero gravity. However, could someone help explain the following: If you take a house on a cold winters day then hot air in the house will rise due to differences in density and some of this air will find some way of escaping out the top of the house. When this warm air escapes, as far as I know, this causes cooler air from outside to be drawn in neat ground level (e.g. under your door). Surely this mechanism is caused by pressure differences? Why else would the cold air come inside the house? If it didn't then you would eventually create a vacuum no? I know it's not possible but if you completely stopped hot air leaving the top of your house would this completely stop drafts at the bottom of the house since a draft is just cold air coming in to replace the escaping hot air? Or vica versa, if you completely blocked cold air coming into your house at ground level would this have the knock on effect of stopping hot air escaping near the top? Thanks to anyone who can help out!


Why isn't the gravitational acceleration taken in to account when determining the net acceleration of a person riding a Ferris wheel?

Say for example you have a person riding a Ferris wheel. What is the difference in his acceleration (towards the center) for when he is at the highest point to when he is at the lowest point? So the answer is that his acceleration is the same every where, that is it is the centripetal acceleration. My question is, why isn't his net acceleration at the top g+Ca, towards the ground, and his acceleration at the bottom g-Ca, towards the sky?


If the multiverse and many-worlds don't exist, how should we interpret probabilities?

If the multiverse of eternal inflation and the many-worlds of quantum mechanics don't exist, how should we interpret the meaning of probabilities? If there is only one copy of the universe out there, what does it mean to say a radioactive nucleus has a 1/2 probability of decaying? Does it mean if we have 1,000,000 similar radioactive nuclei, 500,000 $\pm$ 1,000 of them will decay? How do we make sense of the $\pm$ 1,000 error bar? Thanks


Paper stiffness

As piece of paper is folded and unfolded, the stiffness of the sheet may seem to be greatly increased. To those of you who don't recognise it: take a sheet of A4 paper, grasp one of the edges and move your hand vertically. Now fold the paper in half, unfold it and repeat. Is anyone aware of any formal or heuristic explanation of what changed?


1D Topological insulator with PT symmetry

Assume I have the Hamiltonian for a 1D topological insulators as: $H=sin(P_x) \sigma_x+i \Delta \sigma_{y}+(1-m-cos(P_x)) \sigma_z $ where $m$ is the mass term, $P_x$ is the momentum and $\Delta$ is an extra term to make the Hamiltonian non-Hermitian. If I choose the time reversal operator $T=i \sigma_y K$ and the parity operator $P=\sigma_{y}$, will the Hamiltonian be PT symmetric?


What meaning do changes in the absolute value of Gibbs free energy have in a simple expansion process?

Below is a simple representation of the thermodynamics of a steam turbine. Stream kinetic and potential energy changes are neglected and no other type of non-PV work is done besides shaft work. Enthalpy and entropy are straight from the steam tables. Gibbs free energy is calculated using $G=H-TS$. Outlet pressure was arbitrarily chosen as 400 psig. Steam quality is 1.0 in both inlet and outlet conditions. What physical significance, if any, does the _change_ in Gibbs free energy have? In this example, note that Gibbs free energy _increases_ which seems counterintuitive for a supposedly spontaneous process. Indeed, if I use an isentropic efficiency of 0 in this model to represent a simple throttle valve, I get a _decrease_ in Gibbs free energy to -798 Btu/lb which makes sense as a spontaneous process. I am familiar with the availability approach. I would just like to understand what significance the change in absolute values of Gibbs free energy have, if any. ![Turbine Thermodynamics](http://i.stack.imgur.com/3JpJU.jpg)


Confusion With How Dimensions Work

Form what I understand if you have an equation such as: $$v = v_0 + at$$ then the dimensions must match on both sides i.e. $L/T = L/T$ (which is true in this case), but I have seen equations such as 'position as a function of time' $x(t) = 1 + t^2$, and obviously time is in $T$, but apparently the function gives you position which is $L$... so what happens to $T$ and where does the $L$ come from? I thought dimensions must always match... Also, let us say that you know the time to reach a destination is proportional to distance i.e. double the distance and you get double the time, now this makes sense to me, but as I said earlier I thought that dimensions must always be consistent or else you can not make comparisons in physics, so if you are giving me $L$ (the distance), how can that become $T$ (time) all of a sudden?


x-rays from dark matter accretion into a supermassive black hole?

I understand that accretion of normal matter into a super-massive black hole leads to x-ray emissions. Is the same effect expected to occur for dark matter accretion into a supermassive black hole? Are there any good articles ( _e.g._ on arXiv) I could read to learn more about this?


Math and Wormholes

Hopefully this is the correct forum for this. I felt that Physics Overflow may not be the correct place. I had a student approach me ask me what kinds of mathematics goes into the study of wormholes. She specifically asked whether there is any amount of topology involved, but I'd appreciate it if I could give her a more full answer. So, topology answers are the best but other areas of math would be nice. I am a topology Ph.D. student but know almost zero physics, just in case my background helps with your answer. I didn't know which tag(s) to add. Thanks.


Is it possible that Atomic Electron Probability Density is a result of Heat?

The Schrödinger Equation provides a Probability Density map of the atom. In light of that, are either of the following possible:   1. The orbital/electron cloud converges to a 2d surface without heat (absolute zero)?   2. heat is responsible for the probability density variation from the above smooth surface? I have taken two calculus based physics, and Modern Physics with the Schrödinger equation, Heisenberg Uncertainty Principle, Etc.


How is the Geometric Phase measured in the experiment?

I had read some papers that have mentioned the geometric phase (Berry phase) can be used to detect the quantum phase transitions in a quantum many-body system. My question is: How is it measured in the experiment?


How much power would it take to stop a bullet with a magnet?

If a 7.5 gram copper-jacketed lead bullet (say, a 9x19mm Parabellum) was travelling at 360 m/s, how much power would it take to diamagnetically stop it in the space of one meter? This question comes from discussion about this question on the SF&F SE, about the scene in the _X-Men_ movie where Magneto stops a bullet before it hits someone in the face. I'm aware that a strong enough magnet to cause this effect is going to have a whole heap of other side effects. I'm mainly interested in the ideal case of simply stopping a bullet, rather than the complication of doing it within an inch of someone's face.


Beginner Quantum Mechanics Questions and Theories

So, I recently read the following article: http://www.preposterousuniverse.com/eternitytohere/quantum/ and I have some questions on some of the things it discussed, as well as some theories of my own. ### My Theories   * I also was starting to think about entanglement, and I eventually came up with the consensus that entanglement is when one thing's superposition is dependent on another thing's superposition or (collapsed) wave function.   * If that is correct, then wouldn't every ~~atom~~ elementary particle be entangled with every other elementary particle in the universe because of the fact that two things can't be in the same space. As soon as the wave function of one of the elementary particles collapses, we are given information about every other particle in existence, because we know that all other particles can't occupy that space anymore. If this keeps happening we would get some sort of ripple effect among the particles. Does this make sense or am I completely off my rockers here? Now the way I eventually came to thinking about "quantum observation" is that if information about a thing's position/status is persistent (it survives from at least one _instance_ of time to the next), then it has been observed. It's kinda like the example in the article where if we don't know where the cat is, we can work off the assumption that it's everywhere at once. As soon as information is recorded about that cat's position, that assumption is no longer valid because information has been provided to narrow down the field of possibilities. (Basically Shrödinger's cat, with a quantum twist). It all boils down to human uncertainty, and the equations we work with are tailored to work with the most specific information, even if that information isn't necessarily one value. Again, does that make any sense at all? (I hope so haha) ### Other Info   * Overall, I'm brand new to quantum mechanics, and have almost no physics background. (You'll know why in a moment.) Some of these are probably crazy to anyone who knows what they're talking about, but it's my first try at understanding these concepts. I'd love to get guidance and get better with this topic as I go on.   * I learn best when I'm given concrete examples and analogies, and then am allowed to branch off from there to understand the more abstract topic, once I have the concrete stuff under my belt.   * I'm only a freshman in high school, so I don't speak calculus yet. Please avoid throwing big equations at me, but if you think I'd benefit from seeing them, please bear that in mind and provide some explanation along with them.


6d Massive Gravity

Massive gravity (with a Fierz-Pauli mass) in 4 dimensions is very well- studied, involving exotic phenomena like the vDVZ discontinuity and the Vainshtein effect that all have an elegant and physically transparent explanation in terms of an effective field theory of longitudinal modes, as explained by Arkani-Hamed, Georgi, and Schwartz. Is there any analogous work on six-dimensional massive gravity? (The right mass term would still be of Fierz-Pauli form, but the little group is bigger and so I would expect a more complicated set of longitudinal modes to think about.) EDIT: added a bounty to renew interest.


Minimal strings and topological strings

In http://arxiv.org/abs/hep-th/0206255 Dijkgraaf and Vafa showed that the closed string partition function of the topological B-model on a Calabi-Yau of the form $uv-H(x,y)=0$ coincides with the free energy of a certain matrix model. Then, after taking the double-scaling limit, they get an identification between the B-model partition function and the minimal string partition function. The latter is a minimal model coupled to the Liouville theory, and the equation $H(x,y)=0$ corresponds to what is known as the minimal string Riemann surface (see http://arxiv.org/abs/hep-th/0312170). For the $(p,q)$ minimal model (without any insertions) one gets $H(x,y)=y^p+x^q$. There are two kinds of branes in the Liouville theory: FZZT and ZZ, where the FZZT branes are parametrized (semiclassically) by the points on the Riemann surface $H(x,y)=0$. What are the equivalents of the FZZT and ZZ open string partition functions in the B-model?


Holographic Field Theory

I am trying to read this paper http://arxiv.org/abs/1204.1780 and I don't understand how to get from eqn 91 which is, $$S_{2} = N^{2} \\{V[P^{(1)}_{m}] + (J^{(1)m} - \mathcal{J}^{m})P_{m}^{(1)}\\} -NJ^{(1)2'} tr(\phi^{2}) - N m^{2} tr(\tilde{\phi^{2}}) - N \sum_{n=3}^{\infty}tr(\phi + \tilde{\phi})^{n}).$$ Here $\phi$ is a low energy mode and $\tilde{\phi}$ is the high energy modes. The partition function is $$Z = \int d\phi d\tilde{\phi} dJ^{(1)n}dP_{n}^{(1)} e^{iS_{2}}.$$ I am interested to remove the high energy fields $\tilde{\phi}$ to obtain a low energy effective action for $\phi$. Which is given in the eqns 92 and 93. But this integral is cannot be evaluated, so if someone knows a technique/method, it would great.


Is it possible to derive the brane action in pure supergravity?

The branes that source the RR fields of supergravity are described by the DBI action plus a CS term. I know this only from superstring considerations. Is there a way to find this result without requiring superstring arguments? It should, but I am not sure. I would say that one consider deformations of a static, flat brane, and asking for the deformation to be again solution of sugra equations would impose conditions on the deformations, themselves equivalent to equations of motion derived from the brane action, but I am not sure. Does someone have seen something like this in the literature? Many thanks for your help.


Finding and plotting the current graph from a voltage graph (piecewise function)

Problem description: ![Problem description:](http://i.stack.imgur.com/tHj5i.jpg) My attempt at making the piecewise function: ![enter image description here](http://i.stack.imgur.com/7qsm4.gif) Then, in order to find i(t), I took the derivative of 80x/3 and -80x/3, with respect to x, and multiplied C. Obviously, I'm left with a unknown variable C, so I'm not sure how to proceed from here. Am I supposed to find C first and then take the derivative? (how do I do that? I've been searching the web for this possibility and can't seem to find anything)


An iPhone falling on carpet is fine, is it true?

I heard that an iphone falling on carpet would not cause damage to it, both internal and external. Any physical explanation for this?


Partition Function of a three state particle system

I've just finished studying the partition function of a two-state particle system, where particles can have a 0 energy value or E energy value . That is: ![enter image description here](http://i.stack.imgur.com/LXZiV.png) ![enter image description here](http://i.stack.imgur.com/aTpjV.png) Where $t_j$ is a variable of value 0 or 1. In this case the set $\\{t_j\\}$ determines a microscopic state of the system and at the same time determines the energy of the respective microscopic state. Having this form of the partition function i can easily get a simple expression that will give me thermodynamics properties of the system. My problem is: Having a N localized non interacting atoms in which each atom can be in one of three quantum states characterized by the quantum number k=-1,0,1 ,an atom has the same energy E>0 in k=1 or k=-1 state, and an atom in k=0 state has E=0 energy. What is the partition function of this system ?


What is light, and how can it travel in a vacuum forever in all directions at once without a medium?

I know there are many questions that are similar (maybe identical?). I am not a physicist nor a student - I am just interested in physics and have been watching many physics channels on youtube recently in my spare time. Here is a (nearly) identical question: Why don't electromagnetic waves require a medium? Here is one response to this question : > Well, I would say the electromagnetic field is the medium. I don't understand this - How can something be the medium for itself? The answers to these questions I don't really think address specific points of confusion I am having - so I am going to try to ask the question a little differently How can light (or electromagnetic radiation) travel through a vacuum when there is nothing there to act as a medium, and do so forever in all directions? For example the light coming from a star millions of light years away. With water, for example, it is easy for me to see how a wave can disperse over the surface in all directions and travel for as long as the energy in the wave doesn't dissipate. The molecules in the water act as the medium, and are oscillating and pushing all neighboring molecules causing a domino effect outwards. But what is the light doing? One thought I had is, if light is made of something then it just moves outwards from the point of origin in whatever direction the light was headed. Which leads me to photons... Here is the first paragraph from Wikipedia : http://en.wikipedia.org/wiki/Photon > A photon is an elementary particle, the quantum of light and all other forms > of electromagnetic radiation, and the force carrier for the electromagnetic > force, even when static via virtual photons. The effects of this force are > easily observable at both the microscopic and macroscopic level, because the > photon has no rest mass; this allows for interactions at long distances. > Like all elementary particles, photons are currently best explained by > quantum mechanics and exhibit wave–particle duality, exhibiting properties > of both waves and particles. For example, a single photon may be refracted > by a lens or exhibit wave interference with itself, but also act as a > particle giving a definite result when its position is measured. So... maybe photons make up light, and they are just shooting through space _like_ a wave, but it really isn't a physical wave (like a sound or water wave), it's just "acting like one"? Here is a video from minutephysics : https://www.youtube.com/watch?v=Q_h4IoPJXZw The video talks about the particle-wave duality of an electron. So a single electron can act as a wave, but when it hits something it hits only one place. Is that the same for photons? **Thought Experiment** : Out in the vacuum of space there is a huge hollow sphere 1 light year in radius. Inside is just vaccuum, and at the exact center a flash of light is emitted in all directions from a point source. 1 year later the light hits the sphere at all places at once. That's 12.6 (or 4 times pi) light years squared of surface area to cover! How can there be enough photons to hit every spot inside the surface area of this huge sphere? I understand that very little light would hit this far away, but even if non-zero at least some light is hitting. Or maybe does this depend on how big the flash was? Maybe it doesn't hit every spot? If light really is made of photons and acts like a wave but each photon hits only one place, then it seems to me there must be an infinite amount of photons in that flash of light to hit so many places inside the sphere. But that must be impossible, so there must be a flaw in my reasoning.


Why is a dipole moment called a dipole moment

The General Formula for a moment is the following one: $$ \vec{M} = \vec{r} \times \vec{F} $$ However the formula for a dipole moment is this one: $$ \vec{p} = Q \vec{d} $$ How comes this is still called a "moment"? A charge is not the same as a force, so if you strictly follow the definition it is not a moment, or am I mistaken?


Dropping a ball in a train moving close to the speed of light?

Suppose a train is moving very close to the speed of light, say 0.999c relative to a stationary observer on Earth. Now a stationary observer on Earth will observe clocks on the train to tick slower than usual. Now suppose a boy within the train drops a ball. The stationary observer measures how long it takes for the ball to hit the ground. Will the result simply be $\sqrt{\frac{2s}{g}}$ or will the result be a greater number due to time dilation?


A question about sign in Euclidean path integral

I have a question about the sign in the Euclidean path integral in Polchinski's string theory vol I, p 337. In page 335, Polchinski introduced path integral in Euclidean space > $$ \langle q_f, U| T \left[ \Pi_a \hat{q}(u_a) \right] |q_i, 0 \rangle_E = > \int [dq]^{q_f,U}_{q_i,0} \exp \left[ \int_0^U du L(q, i \partial_u q) > \right] \Pi_a q(u_a) \tag{A.1.30} $$ The weight in the Euclidean path > inegral is conventionally defined to be $\exp(-S_E)$, so the Euclidean > action contains a minus sign $$ S_E = \int du L_E(q, \partial_u q) = - \int > du L(q, i \partial_u q) \tag{A.1.31} $$ If I used the sentence above (A.1.31) and (A.1.31), I got $\exp(-S_E)=\exp(\int du L(q, i \partial_i q))$ Later in page 337, it is said > Begin with the free action $$ \frac{1}{2} \int d^d \phi(x) \Delta \phi(x) > \tag{A.1.38}$$ ... $$ Z[J] = \left\langle \exp\left[ i \int d^d J(x) \phi(x) > \right] \right\rangle $$ $$ = \int [d\phi] \exp \left[ -\frac{1}{2} \int > d^dx \phi(x) \Delta \phi(x) + i \int d^d x J(x) \phi(x) \tag{A.1.39} > \right]$$ It seems the action (A.1.38) is in Euclidean space, otherwise the minus sign in (A.1.39) won't appear. However, Polchinski's book is using $(-+++)$ metric. Under this metric, the action free scalar field is $$\int d^dx - \frac{1}{2} \partial_{\mu} \phi \partial^{\mu} \phi -\frac{1}{2} m^2 \phi = \int d^dx + \frac{1}{2} \phi \partial^{\mu} \partial_{\mu} \phi -\frac{1}{2} m^2 \phi $$ I got a **plus** sign in front of the kinetic term after integrating over variables. In (A.1.39) it is a **minus** sign. My question is, why I got a different sign?


Dimension of the space of solutions in an electric circuit

Consider an electric circuit with dc sources ( voltage and current) and resistors. Write down the equations. In the most general case, the solution of the system is not unique. The set of solutions can be empty or positive dimensional (simple example: 2 points in the graph and two batteries in parallel joining the points). The dimension of the space of solutions can be computed with two different methods: 1) Mathematically : compute the determinants and the compatibility conditions 2) Physically. Give conditions on the graph to give a meaning for the dimension. For instance a cycle of batteries give an empty set of solutions or make the dimension increase by one. I am interested in second method and I am looking for references in the litterature. What I found is the following. In the student books, the unicity is always assumed to be true, eg. in the standard book by nilsson and riedel. In more advanced books, I have found discusions only for particular cases, and with very technical tools. For instance, in Frankel (The geometry of physics), only purely resistive circuits are considered with source of currents in the nodes. And the proof uses ( a simplified version of) Hodge theory. Now my question: \- Is there a book or an article where the very general case ( any graph with dc sources and resistors ) is considered and the dimension of the system described in terms of the graph ? I am interested both in sophisticated answers as above and in answers with basic tools of linear algebra. All references welcome. I am looking for references, not for the solution. I have already written a solution for my students (an elementary one, with basic linear algebra). I want to compare my solution with the existing litterature. If this is useful and not a waste of time, I will make public my personal notes.


Radius of curvature of lens: Newtons Ring method vs spherometer

I have to do an experiment to find out the radius of curvature of a lens using the Newtons ring method given that you know the wavelength of the monochromatic light used in the experiment. What the question is that why we are using the Newtons ring when we have a spherometer available which expects the answer that the measurement by one of them is more accurate than the other( Internet tells me that the newtons ring is more accurate than the spherometer without citing a reason) So in short what I want to know about is **_Why is using Newton's Ring method to calculate the radius of curvature of the lens more accurate than the spherometer measurement_**. Some of my observation so far   1. I have compared the measurements of the radius of curvature of my friends measurement, and found that the newtons ring and spherometer measurement deviates upto 3cm.   2. When I have looked at the principle behind both of the experiments I have found no approximation in the formulas making the question puzzling


A clock in freefall

Time dilation calculated using Schwarzschild metric for a non rotating spherical body is: $$t_0=t_f\sqrt{1-\frac{2GM}{rc^2}}$$ For such a non rotating spherical body, what would the time dilation of a clock in vacuum free-falling from infinity be? (If the answer is non-trivial; a high level outline of the calculation would suffice / be appreciated) **Edit:** I am currently working on an iOS app that is trying to model the mechanism underpinning relativity. So, far the mechanism that I have created is shockingly simple and shockingly good at conforming to Relativity. However, I am trying to break it. I am trying to find any possible areas where the two may diverge. I have noted that using my model a clock in freefall will experience no time dilation, i.e. $t_0=t_f$ and I want to make sure Relativity agrees. I have noted the gravitational component of time dilation above. Since my clock is moving one might also expect a kinematic time dilation. I can calculate the velocity of my clock: $$E_k=\frac{1}{2}mv^2$$ $$E_p=\frac{-GMm}{r}$$ $$v=\sqrt{\frac{2GM}{r}}$$ Plugging this velocity into the kinematic time dilation equation: $$\Delta t'=\frac{\Delta t}{\sqrt{1-\frac{v^2}{c^2}}}$$ $$\Delta t'=\frac{\Delta t}{\sqrt{1-\frac{2GM}{rc^2}}}$$ At this point one might make the observation that the kinematic dilation is the inverse of the gravitational dilation and therefore conclude that: $$t_0=t_f$$


water in vacuum(or space) and temperature in space

So, water in vacuum will boil first and then freeze. I don't know how the freeze happens. As pressure lowers to zero, what happened to freezing point?(I know heat taken by vapor, and the water cool down, but I don't think it will be that cold, will it? In vacuum, boiling point is so low that water shouldn't need so much heat as it does in normal pressure, which means vapor actually takes more heat away under normal pressure than in vacuum, so water under normal pressure would be cooler??(I'm guessing) And temperature comes from heat generates by motion of molecules(I guess so), so in vacuum, there is no temp??? What happen when I heat up a vacuum tube? Does heat need a medium to “travel"?


I learned that a moving charge creates a magnetic field perpendicular to its direction of motion. I also learned that charged particles like electrons have spin and they also create a magnetic field because of their magnetic dipole moment. I don't understand what magnetic dipole moment is, and I can't find a decent explanation for it by using Google. First, Can someone please explain magnetic dipole moment arising from spin in an effective way? Second, is my main question; Does a magnetic field arise from a moving charge or from its spin, or both? Third, I am very confused. I think I am mixing together the classical electrodynamics description of magnetic field, (moving charge), with quantum mechanics description of magnetic field, (magnetic dipole moment due to spin?)


Observing visible light itself by our eyes

We know that for any object to be seen by a human eye, the light from any source must be reflected from that object so that it reaches our eye and we can detect it.....but how do we see light itself? Bcoz obviously light can't bounce off light itself, it just pass right through each other, do our eyes absorb the light for it to be detected??


What is the official difference between a planet and a dwarf planet?

I'm trying to understand how objects are classified as planets, moons, or dwarf planets. Can someone please explain the differences between them? I'm really curious about why Pluto is a dwarf planet, for example.


Population I and II stars

I have been thinking about the formation of the galaxy. I can easily understand that old, low-metallicity stars are in the halo, but I'm missing something when it comes to the disk. If slowly-rotating gas settled into the disk, formed stars, then did the disk at one time consist of Population II stars which have since exploded? Why did not the same processes happen in both the disk and the halo? That would mean that a lot of Population II halo stars will have exploded and formed plenty of Population I halo stars . . . but without disk dynamics perhaps not many would form. How about the missing Population II disk stars, why are they not still around like the halo clusters?


Photon wave packets from distant stars

A distant star like the sun, thousands of light years away, could be so faint that only one photon might arrive per square meter every few hundred seconds. How can we think about such an arriving photon in wave packet terms? Years ago, in a popularisation entitled “Quantum Reality”, Nick Herbert suggested that the photon probability density function in such a case would be a macroscopic entity, something like a pancake with a diameter of metres in the direction transverse to motion, but very thin. (I know the wave packet is a mathematical construct, not a physical entity). I have never understood how such a calculation could have been derived. After such a lengthy trip, tight lateral localisation suggests a broad transverse momentum spectrum. And since we know the photon’s velocity is c, the reason for any particular pancake “thickness” in the direction of motion seems rather obscure. (Herbert then linked the wave packet width to the possibilities of stellar optical interferometry).


Is the "pressure" as we know a kind of "instantaneous action at a distance"?

This may be a very simple question, but I still dont know how that work in the essence (quantic/atomic level) Consider the image bellow: ![enter image description here](http://i.stack.imgur.com/qcggt.png) How the "pressure (air atoms) " known that in the other side of the barrier there is nothing and pull the bairrer to the "low pressure". There is some sort of instantanious action at a distance to the particles knowing where are holes to go ? Where does the pressure comes from ? In a practical way, how the water in the ocean know that there is air inside the iron ship to pull it upwards out of water ?


Internal and spacetime symmetries?

I am trying to understand Wiki's explanation about correlation. Part of this article talks about internal and spacetime symteries: > If the probability distribution has any target space symmetries, i.e. > symmetries in the space of the stochastic variable (also called internal > symmetries), then the correlation matrix will have induced symmetries. If > there are symmetries of the space (or time) in which the random variables > exist (also called spacetime symmetries) then the correlation matrix will > have special properties. And then it follows with two examples about symmetries. However I didn't quite understand the definition nor I know any intuitive example for each case? Can somebody help me regarding this; first an explanation of definition and secondly some examples. What does it mean to say "target space" symmetries in space of stochastic variable and what is the difference of that to "symmetries of the space (or time) in which the random variables exist"?


comparing torque and 0-60 speed in cars

Im looking a two cars, both are the same model but ones got a 2.0L turbo petrol engine and ones got a 3.0L turbo diesel engine. These are the full specs **2.0L Petrol** ![enter image description here](http://i.stack.imgur.com/herd0.png) **3.0L Diesel** ![enter image description here](http://i.stack.imgur.com/HopJW.png) what im wandering is that although the 0-60 speed is similar the torque is almost 50% more. Ive driven the diesel and what i really liked about its 'effortless' acceleration, is this down to the torque, rather than the speed - if that makes sense ? sorry if thats a very obvious question my knowledge of physics is 0.


How can I get average numbers of days wind flow from 16 direction?

Wind flow from North direction a nos. of days, Similarly from North East b nos. of days, East c nos. of days, South East d nos. of days, South e nos. of days, South West f nos. of days, West g nos. of days, North West h nos. of days. Then how can I get average numbers of days wind flow from 16 directions ? 16 directions are N,NNE, NE, NEE, E, SEE,SE, SSE,S, SSW,SW, SWW, W, NWW, NW, NNW.


What are renormalons from a physics point of view?

This is again a question in the context of this paper about the Exact Renormalization Group. On p 23 and the following few pages, it is explained that for a $\lambda \phi^4$ bare action at the bare scale $\Lambda_0$, after integrating out degrees of freedom and assuming a small coupling $\lambda$, the effective action at the larger scale $\Lambda$ can be written as the sum of a perturbative series plus nonperturbative power corrections (Eq. 2.6) $$ S_{\Lambda,\Lambda_0}[\phi] = \sum\limits_{i=0}^{\infty}\lambda^{i-1}S_i[\phi] +O(\Lambda/\Lambda_0) $$ Naively taking $\Lambda_0 \rightarrow \infty$ makes the second part disappear and leaves the first part which is self-similar which means that the theory is renormalizable. However, as stated in the paper, UV renormalons can make the perturbation series ambiguous, such that the power corrections and therefore the bare scale can not be removed and the theory would then not be renormalizable. The issue is explained by the mathematical argument that in order to obtain a unique finite value for the perturbative series, one makes use of the so- called Borel transformation to define a function which has the same power series expansion defined as an integral in the complex Borel plane. This integral exists only of there are no poles on the real axis, otherwise the contour of integration around these poles can be deformed which makes the integral and the function defined by it ambiguous. If this is the case, the power correction terms have to be kept to restore the uniqueness of the integral which means that the bare scale can not be removed. In the paper it is said, that the poles on the real axis are for example due to UV renormalons, that arise from large loop momenta in certain Feynman diagrams. My question now is: What are these renormalons from a physics point of view? How do they enter the Lagrangian of the theory? Are they some kind of unphysical auxilliary fields that appear in the mentioned Feynman diagrams? And what do the Feynman diagrams that contain them look like?


Quantizing first-class constraints for open algebras: can Hermiticity and noncommutativity coexist?

An open algebra for a collection of first-class constraints, $G_a$, $a=1,\cdots, r$, is given by the Poisson bracket $\\{ G_a, G_b \\} = {f_{ab}}^c[\phi] G_c$ classically, where the structure constants are functions of the dynamical degrees of freedom, $\phi$. When quantizing a gauge theory, a physical state $|\psi\rangle$ has to satisfy the first-class constraints $\widehat{G}_a |\psi\rangle = 0$. From this, one can easily see $[\widehat{G}_a, \widehat{G}_b]|\psi\rangle = 0$. In the quantum version of the theory, the Poisson equation has to be replaced by an operator commutator equation. In general, ${\widehat{f}_{ab}^c}[\widehat{\phi}]$ doesn't commute with $\widehat{G}_c$. One possibility is the right hand side of the equation for the commutator of two constraints is ordered so that the constraint $\widehat{G}_c$ is always on the right in the operator product. However, the resulting product will be nonhermitian in general due to noncommutativity. The commutator of two Hermitian operators is always antihermitian. So, this means the first-class constraint operators have to be nonhermitian. If we want the constraint operators to be hermitian, we require $[\widehat{G}_a, \widehat{G}_b] = i O(\widehat{f}_{ab}{}^c[\widehat{\phi}]\widehat{G}_c)$ where $O$ is some form of operator ordering. However, this operator ordering will in general contain some terms which don't annihilate $|\psi\rangle$ in general because $\widehat{G}_c$ won't always be on the right. How does one get around this?


Determining the center of mass of a cone

I'm having some trouble with a simple classical mechanics problem, where I need to calculate the center of mass of a cone whose base radius is $a$ and height $h$..! I know the required equation. But, I think that I may be making a mistake either with my integral bounds or that $r$ at the last..! $$z_{cm} = \frac{1}{M}\int_0^h \int_0^{2\pi} \int_0^{a(1-z/h)} r \cdot r \:dr d\phi dz$$ 'Cause, once I work this out, I obtain $a \over 2$ instead of $h \over 4$...! Could someone help me?


Why so much geometry in principia and others

I was recently reading 'On the the shoulders of giants' by stephen hawking and looked at many physics(mechanics) problems solved by Copernicus,Newton etc. why is there so much of geometry used by them. I mean, most of the problems are solved using pure math and not much physics. Why is this?


Why are Lagrangian subspaces called 'Lagrangian'?

I am wondering what the special role of Lagrangian subspaces (or submanifolds) are in mechanics. Do these subspaces have some sort of special property for which we have some sort of `Lagrangian property?' The definition I know is, a subspace $Y$ ( submanifold $Y\ $) of a symplectic vector space $(V,\omega)$ (symplectic manifold $(M,\omega)$) is called Lagrangian if $\omega|_{Y\times Y}\equiv 0$ and $\dim Y=\frac{1}{2}\dim V$.


Could we tell the difference between population I and II (or even III) neutron stars?

This question is related to thoughts I was having about the mass-radius relationship for neutron stars. Is it unique? Is there a single relationship between $P$ and $\rho$ or is there any chance that the NS "remembers" the composition or type of star from which it was born? So, suppose I were able to measure the masses and radii of a number of (isolated) neutron stars in the solar neighbourhood. Most of these would presumably be population I, but would I be able to spot a population II/III interloper?


Exception to weight of hot vs cold water?

I have seen the various posts regarding the comparison of weight of heated and cold water.But is there any contradiction?I live in really hot conditions and as such tap water literally 'boils' here.What we do is to let the water flow to release the initial hot water and later obtain the cold water. Now , as water storage tanks have the supply ducts at the bottom and the hot water at the top,how is it that one gets hot water first? Could someone please ,explain this?


Is it possible to derive Schrodinger equation in this way?

Let's have wave-function $\lvert \psi \rangle$. The full probability is equal to one: $$\langle \Psi\lvert\Psi \rangle = 1.\tag{1}$$ We need to introduce time evolution of $\Psi $; we know it in the initial moment of time. So it's naturally set $$\lvert \Psi (t) \rangle = \hat {U}|\Psi (0) \rangle ,$$ and from $(1)$ it follows that $$\hat {U}^{\dagger}\hat {U} = \hat {\mathbf E}.$$ So it may be represented as $U = e^{i \alpha \hat {H}t}$ (we suppose that $\hat {H}^{\dagger} = \hat {H}$ and $\hat {H} \neq \hat {H}(t)$ for simplifying derivation). Thus it is possible to write $$\partial_{t}\lvert\Psi (t) \rangle = i\alpha \hat {H}| \Psi\rangle.$$ But how to get the physical interpretation for $\hat {H}$?


How to prove equations for energy of a wave

In my textbook, it says that the energy (I) of a wave, determined by the power of a wave (P) divided by area (A), is determined by the following formula: $$I = \frac{1}{2}\rho v\omega^2A^2 = 2\pi^2 \rho vf^2A^2$$ where $\rho$ is density, $v$ is propagation speed, $\omega$ is angular speed ($\omega = 2\pi f$ and $A$ amplitude. Can anyone show how it is like this?


Calculating the required amount of acid/base to reach a pH

Bit stuck on working this out - or if it is even possible. I have a reservoir of solution (water/plant feed/dead flies) that is pH 6 and I want to get it to pH 6.25. Here is the information I have:   * Size of reservoir: 50 liters   * pH of reservoir: 6   * Required pH: 6.25   * Alkali/Base Stuff: Potassium Hydroxide at 50% Strength Is there any way I can work this out to say, 15 or 20%? Many Thanks


Electromagnetic-Radiation

1. describe how radiation from a resonant LC circuit is increased by opening out the capacitor plates?   2. Explain that a straight wire provides reactive elements of a LC circuit.


Freshman year forces problem

I have been sitting with this problem for about 2 hours now. I know it involves simultaneous equations of some sort... I am trying to figure out maybe if there was not enough information given .. maybe the community could help me out :) Question: _It Is Question 2_ ![Image of question 2](http://i.stack.imgur.com/UDhJC.png) My Approach   ![enter image description here](http://i.stack.imgur.com/Q6QNS.png)


Conflict between Lippmann–Schwinger equation and Gell-Mann and Low theorem about energy

Lippmann–Schwinger equation states that scattering state will have the same energy as free state, while Gell-Mann Low theorem says that they have different enery. Lippmann–Schwinger equation says: That is, if $|f\rangle $ is the eigenstate of interacting picture's free Hamiltonian $\hat{H}_0^{(I)}$ with eigenvalue $E$, then $$|F \rangle := \hat{U}(0,-\infty)|f\rangle$$ is the eigenstale of the full Hamiltonian $\hat{H}^{(I)}(0)$ at $t=0$ with the same eigenvalue $E$, where $\hat{U}(t,t_0)$ is the evolution operator in interacting picture. On the other hand, Gell-Mann and Low theorem states that : Let $|f\rangle$ be an eigenstate of $H_0$ with energy $E_0$ and let the 'interacting' Hamiltonian be $H=H_0 + gV$, where $g$ is a coupling constant and $V$ the interaction term. We define a Hamiltonian $H_\epsilon=H_0 + e^{-\epsilon |t|}gV$ which effectively interpolates between $H$ and $H_0$ in the limit $\epsilon \rightarrow 0^+$ and $|t|\rightarrow\infty$. Let $U_{\epsilon I}$ denote the evolution operator in the interaction picture. The Gell-Mann and Low theorem asserts that if the limit as $\epsilon\rightarrow 0^+$ of $|F_\epsilon \rangle = \frac{ U_{\epsilon I} (0,-\infty) |f\rangle}{\langle f | U_{\epsilon I}(0,-\infty)|f\rangle}$ exists, then $|F_\epsilon \rangle$ are eigenstates of $H$. But the energy $E$ will not be same as $E_0$ and will have a shift. Here is a proof of my first state that energy will be same ![enter image description here](http://i.stack.imgur.com/rhih5.png) ![enter image description here](http://i.stack.imgur.com/rH5wk.png) ![enter image description here](http://i.stack.imgur.com/onN6Q.png) ![enter image description here](http://i.stack.imgur.com/zbIlB.png) ![enter image description here](http://i.stack.imgur.com/QZ3RB.png) ![enter image description here](http://i.stack.imgur.com/ArOis.png) ![enter image description here](http://i.stack.imgur.com/9kv4n.png) ![enter image description here](http://i.stack.imgur.com/7TjRL.png) ![enter image description here](http://i.stack.imgur.com/GkYAw.png)


“Relativistic Baseball”

On Randall Munroe’s new blog “what if”, he answers the question: > “What would happen if you tried to hit a baseball pitched at 90% the speed > of light?” http://what-if.xkcd.com/1/ He concludes: > … the air molecules in front of this ball don’t have time to be jostled out > of the way. The ball smacks into them hard that the atoms in the air > molecules actually fuse with the atoms in the ball’s surface. Each collision > releases a burst of gamma rays and scattered particles. > > These gamma rays and debris expand outward in a bubble centered on the > pitcher’s mound. They start to tear apart the molecules in the air, ripping > the electrons from the nuclei and turning the air in the stadium into an > expanding bubble of incandescent plasma. > > … Suppose you’re watching from a hilltop outside the city. The first thing > you see is a blinding light, far outshining the sun. This gradually fades > over the course of a few seconds, and a growing fireball rises into a > mushroom cloud. Then, with a great roar, the blast wave arrives, tearing up > trees and shredding houses. > > Everything within roughly a mile of the park is leveled, and a firestorm > engulfs the surrounding city. The baseball diamond is now a sizable crater, > centered a few hundred feet behind the former location of the backstop. Is that all actually reasonable, or is it all just hyperbole?


Capillary tube of insufficient length

I was wondering if we have a very thin glass tube placed in a tub of liquid and the portion of the tube outside the liquid is lesser than the height to which the liquid can rise because of capillarity, what will happen? Will the liquid overflow?