Phase velocity is defined as $v_p=\frac{\omega}{k}$ and is described in various textbooks as being the speed at which the phase of a wave propagates. If you have a wave train that is modulated by an envelope, then while the group velocity gives you the speed of the envelope the phase velocity gives you the speed of the wave within the envelope.  Do $v_p$ and $v_g$ have any meaning if we are only considering one sinusoidal wave and not a superposition of such waves? I have been told that the speed of a single sinusoidal wave is its phase velocity, and that phase velocity is what we're really referring to when we talk about a wave's speed.   However, how can phase velocity be greater than light if this is the case? The standard response seems to be that the group velocity is what determines the rate of information transfer, and it is this velocity that cannot exceed $c$. However, in special relativity a simple limit is set on the speed of any particle, and photons must travel only at speed $c$. If $v_p&gt;c$ then does this not mean that the physical wave within the envelope is superluminal, and so the photons that comprise it are also superluminal?
104305
I understand that information cannot be transmitted at a velocity greater than speed of light. I think of this in terms of the radio broadcast: the station sends out carrier frequencies $\omega_c$ but the actual content is carried in the modulated wave, the sidebands $\omega_c\pm\omega_m$. The modulation envelop has its group velocity and this is the speed at which information is being transmitted. We also know, for example, that x-rays in glass have a phase velocity which is greater than the speed of light in vacuum.   My question is, what exactly is it that is travelling faster than the speed of light?    EDIT: I know it couldn't be anything physical as its superluminal. My problem is, what is it that has absolutely no information content yet we associate a velocity with it.
103576
It makes intuitive sense to me for the apparent frequency of a sound as modified by the Doppler effect to be based entirely on the speed at which the observer and the source get closer or farther apart, regardless of whether the source or the observer is moving, based on the principle of relativity. However, the general equation I receive from every source $$f'=f((v±v_o)/(v∓v_s))$$ does not have this property. For example, if the speed of sound in the medium is 2 units and the observer and source are coming closer at a rate of 1 unit, then if we take the observer to be moving then we get $$f'=f(3/2)$$ but if we take the source to be moving then we get $$f'=f(2/1)$$ What is the difference between these two cases?
93328
In some media, mass-carrying particles can go faster than light:     Cherenkov radiation, ... is electromagnetic radiation emitted when a charged particle... passes through a dielectric medium at a speed greater than the phase velocity of light in that medium   How does this happen? I would imagine that light faces the least resistance to motion and that other charged mass-carriers have more stuff to impede their speed.  Why are the particles allowed to exceed the speed of light for other media when they are not allowed to in a vacuum?
61414
I'm just a layman with no real science background  (a) I'm not really sure what a shadow is. Some people are saying that it's the absence of something (a photon?) and therefore it is not subject to laws of physics. But I know when I'm in a shadow and when in sunlight... that's "information", and I thought that information flow can't break the speed of light limit.  (b) Some others use the laser pointer/flicking wrist example to talk about laser dots moving faster than light speed. But that can't be right, can it? The photons from the laser point have to travel from A to B and (after wrist flick) from A to C ... but the photons are all moving at the speed of light - there's no law-breaking in moving the laser dot from point B to point C because the photons are moving either from A to B or A to C ... the distance from B to C is not material (is it?).  (c) Also, can a shadow be thought of as a gap in a photon flow? E.g. if I interrupt a beam of single photons then can I think of it as e.g. PPPSSSPPPSSSPPPSSSPPP, etc where P = photons and S(shadow) = no photon &amp;, if so, the "SSS" is effectively travelling at the speed of light ... no faster.
64188
A while back in my Dynamics &amp; Relativity lectures my lecturer mentioned that an object need not be accelerating relative to anything - he said it makes sense for an object to just be accelerating. Now, to me (and to my supervisor for this course), this sounds a little weird. An object's velocity is relative to the frame you're observing it in/from (right?), so where does this 'relativeness' go when we differentiate?   I am pretty sure that I'm just confused here or that I've somehow misheard/misunderstood the lecturer, so can someone please explain this to me.
15718
In replying to a  I stated:     Gravitational waves have not been yet experimentally observed so as to have their velocity measured.   Which after the fact prompted me to try and verify it. I found a "speed of gravity" claimed measurement, which has been in a controversial discussion. In the last paragraph of  I found:     In September 2002, Sergei Kopeikin and Edward Fomalont announced that they had made an indirect measurement of the speed of gravity, using their data from VLBI measurement of the retarded position of Jupiter on its orbit during Jupiter's transit across the line-of-sight of the bright radio source quasar QSO J0842+1835. Kopeikin and Fomalont concluded that the speed of gravity is between 0.8 and 1.2 times the speed of light, which would be fully consistent with the theoretical prediction of general relativity that the speed of gravity is exactly the same as the speed of light.[18]   There has been criticism, described in the paragraph.   Has this matter been resolved ? is the measurement valid?  The speed of gravity has also been  calculated from observations of the orbital decay rate of binary pulsars and found to be consistent to the speed of light within 1% ( same link).  This is the speed that gravitational fields transfer information, and though the number itself depends on the theoretical framework solving for the system parameters it is true that just the existence  of gravitational damping in the binary system would imply that the speed cannot be infinite.  Along these lines I was wondering whether studying the second star in a binary system where the first has gone nova would not give a cleaner and model independent measurement of the speed of transfer of information gravitationally. In this day and age where everything is digital one should be able to have data on this from now on.     (Feb 11,2016) announced the observation of gravitational waves , at the same time  consistent with the speed of light being c, and thus measuring it. See the  for a summary.
31539
I have a volume with $N$ molecules; I need to assign to each particle a velocity vector: $$|\mathbf{v}_{i}|=[v_{x}, v_{y}, v_{z}]^{T}$$ for the $i$-th molecule; the velocities must follow the Maxwell-Boltzmann distribution; here is the question: I know the definition of the M.B. distribution, how do I extract the velocities that I have to assign to each molecule? Basically what I want to know is how to calculate $$v_{x}=?$$ $$v_{y}=?$$ $$v_{z}=?$$ for each molecule.
66800
There is a problem with my logic and I cannot seem to point out where. There's a rocket ship travelling at close-to-c speed v without any acceleration (hypothetically), and there is an observer AA with a clock A on Earth, and there's another observer on the rocket BB with a clock B and these two clocks were initially in sync when the rocket was at rest with a FoR (frame of reference) attached to Earth. Now, this rocket's moving and AA tells us that B is running slower than A by a factor of$$ \gamma = 1 /(1-v^2/c^2)^{1/2} $$ $$ t_A/t_B = \gamma $$ where v is the relative 1d- velocity between the two i.e the earth and the rocket! That would mean the time elapsed on A is greater than that on B but this will happen only in the FoR of AA? So $t_B$ in this equation must be the time on B as observed by AA? Is this correct? What do the terms mean in the equations? If the symmetry holds and BB doesn't accelerate, then BB could say that $$ t_B/t_A = \gamma $$ right? where $t_B$ and $t_A$ are the times on B and A with respect to FoR of BB? But I was solving this problem and I took the earth FoR of A, but the prof took the rocket FoR of B? Like how will I know which FoR to solve the problem from? It'd greatly help if the terms in all the above equations were laid down neatly! DO we even need these FoRs?? Because in all the solved problems the prof is't specifying any and is using random ones! Please help!!!! This is the question where I messed up. The first rocket bound for Alpha Centauri leaves Earth at a velocity (3/5)c. To commemorate the ten year anniversary of the launch, the nations of Earth hold a grand celebration in which they shoot a powerful laser, shaped like a peace sign, toward the ship.  According to Earth clocks, how long after the launch (of the rocket) does the rocket crew first see the celebratory laser light?  This must be 25 years. My reasoning is: If v = 3c/5 10v + vt = ct where t is time taken by the light to reach the rocket from Earth as calculated from Earth.. and I solved that for t. and added 10 years to that because the time starts at the launch of the rocket!  According to clocks on the rocket, how long after the launch does the rocket crew first see the celebratory laser light?  This is 20 years. Here, I say: If it takes 25 years as observed by clocks on earth for the laser to reach the rocket, what should be the corresponding time as seen on a clock on the rocket? Using the formula: 25 = $\gamma$t where $\gamma$= 5/4 solved for t!  According to the rocket crew, how many years had elapsed on the rocket's clocks when the nations of Earth held the celebration? That is, based on the rocket crews' post-processing to determine when the events responsible for their observations took place, how many years have passed on the rocket's clocks when the nations of Earth hold the celebration?  For this, I did the following: 10 years on earth = T years on rocket ship where T must be lesser than 10 as observed from Earth FoR! Therefore, T= 4(10)/5 years = 8 years! But, prof says, 10 years on Earth = T years in rocket ship where T must be GREATER than 10 as observed from the rocket FoR??? Therefore, T = 10(5/4) years = 12.5 years!! What does this question actually want?
55552
I know that as the velocity increases, the mass of the object also increases so it becomes tougher and tougher to move the object which ultimately leads to a requirement of infinite energy to accelerate an object to the speed of light. But I have a doubt.  As far as I know only the observable mass (Relativistic mass) increases but not the Proper Mass or Intrinsic Mass, right? The actual mass of the object will remain the same. So if the actual mass ($m_0)$ remains the same and only the observable mass increases, why is more and more energy required? I know that even if the object exceeds the speed of light, we will not be able to say that it is moving faster than the speed of light but is it possible to make it move faster than light?  If rest mass does not change with $v$ then why is infinite energy required to accelerate an object to the speed of light?  I quote Igor Ivanov (a stackexchange user from this question ) to give an insight into my question.     The mass (the true mass which physicists actually deal with when they calculate something concerning relativistic particles) does not change with velocity. The mass (the true mass!) is an intrinsic property of a body, and it does not depends on the observer's frame of reference. I strongly suggest to  popular article by Lev Okun, where he calls the concept of relativistic mass a "pedagogical virus".   EDIT  After reading the answers I have a doubt.  SO can I say that only the overall energy of the system increases while the mass remains constant? But then if mass remains constant, then why is more and more energy gradually required? I mean there should be a reason for that requirement of infinite energy.
108697
I am very new to relativity, and as I was going through Resnick &amp; Halliday, I saw that momentum was no longer $mv$, rather $\gamma mv$. This was the proof:     $$p =  mv = m \frac{\delta x}{\delta t_0}  $$   where $\delta t_0$ is the proper infinitesimal time interval to cover $\delta x$ distance.     Using the time dilation formula, $\delta t = \gamma \delta t_0$ we can then write    $$ p = m \frac{\delta x}{\delta t_0} = m \frac{\delta x}{\delta t} \frac{\delta t}{\delta t_0} = \gamma m \frac{\delta x}{\delta t} $$   However, since $ \frac{\delta x}{\delta t} $ is just the [observed] particle velocity,    $$ p = \gamma mv $$   On the other hand, when I went on to Feynmann's book I saw that there he talked about using "the new m" (the relativistic mass) to derive all the equations and said      $$p = mv = \frac{m_0 v}{(1 - \frac{v^2}{c^2})} = \gamma m_0 v $$   My problem is that if both these proofs are true, then why doesn't one consider the other's argument, id est, why aren't both of these factors considered together ? If the mass is really changing, why doesn't Resnick &amp; Halliday account for it? That way the momentum would turn out to be $p = \gamma^2 mv$  Am I missing something here?
37234
Maybe an illegal question. Sorry.  Having read that a photon has both wave and a light characteristics it is natural for a novice like myself to wonder "what is the amplitude? "  Naturally I have looked on wiki and other sources and there are a number of answers some of which appear not so much to contradict one another of varying opinions of which is the most meaningful in physics.   I seem to get a higher number of physics sources saying the amplitude depends on how many photons you have and if you have one the question has no meaning.  Of course I have not idea if this is accurate or how one would even test this.   Maybe there are some physics teachers in the audience that can give me a layman's intuition with a little math tossed in.
63732
I am reading Berkeley Physics Course vol. 4 (Quantum Mechanics) , chapter 4 (photons). (1)  Section 46: book says: consider a typical photon emitted by  the source. It can be regarded as a a wave train of finite duration, spreading out in all directions in space and carrying a total energy $\hbar\omega$. If this is true, a photon emitted by a star which is few light years away, would have spread over a sphere of radius of few light years. Then it is not possible to &quot;detect&quot; that photon using a detector.  Then  why does the book say like this?  In section 47, the book gives same example of photon spreading over light years. But in this section, it says that how the people can use this as a paradox; and clarifies that paradox by saying that classical expression for &quot;energy density&quot; refers to a large number of photons, and not for single photon.  So does photon really spread in all directions?   (2)  In Section 38 book says: almost monochromatic photons can not be split into 2 photons of the same frequency which carry only a fraction of the energy of the original photon. While in section 48, it says (with reference to double-slit diffraction experiment) photon came through BOTH the slits.  Does a photon REALLY goes through both the slits? Doesn't it indicate that a single photon can be split into two photons of same frequency ?   Any help will be appreciated. Thanks
107331
Waves are generally classified as either transverse or longitudinal depending on the they way the propagated quantity is oriented with respect to the direction of propagation. Then what is a ? It doesn't make sense to me that a disturbance in the curvature of spacetime has a "direction", so I would say they're neither, much like a wave packet in quantum mechanics.
40878
The Maxwell-Boltzmann velocity distribution in 3D space is $$ f(v)dv = 4\pi \left(\frac{m}{2\pi k_B T}\right)^{3/2} v^2 \exp\left(-\frac{m v^2}{2k_B T}\right)dv$$ It gives the probability for a single particle to have a speed in the intervall $[v,v+dv]$. But this probability is not zero for speeds $v &gt; c $ in conflict with special relativity.  Is there a generalization of the Maxwell-Boltzmann velocity distribution which is valid also in the relativistic regime so that $f(v) = 0$ for $v&gt;c$ ? And how can it be derived? Or can a single particle distribution simply not exist for relativistic speeds, because for high energies, we always have pair-production meaning the particle number is not conserved and a single particle distribution can not be defined in a consistent way?
83328
From this ,     Let’s say Alice is holding a light clock, and Bob is watching her run by, while holding it, with speed V.  Alice is standing still (according to Alice), and the time,$ \tau$, between ticks is easy to figure out: it’s just $\tau = \frac{d}{C}$.  From Bob’s perspective the photon in the clock doesn’t just travel up and down, it must also travel sideways, to keep up with Alice. The additional sideways motion means that the photon has to cover a greater distance, and since it travels at a fixed speed (EEP y’all!) it must take more time   My question is, why does photon need to travel sideways to hit the Alice light clock?
56665
My question is not about why speed of light has a particular constant value which has been sufficiently addressed in other questions on SE already. I want to know whether the fact that speed of light has to be constant is a consequence of other fundamental laws of Universe, or is it just a fundamental law itself.  My motivation for asking this question is thinking about Special Theory of Relativity. One way of understanding why time dilation and length contraction happen is to look at it from the perspective of constancy of speed of light. Alice who is on the moving source of light and Bob, a fixed observer see the speed of light as constant. In order for Alice to see the speed of light as a constant value, time slows down and length expands (in forward direction) thus keeping the speed constant.   In the  I'm doing, the Professor tells that this constancy of speed of light could be more generally referred to as, "Principle of wave constancy". That is, the speed of any wave is a property of the medium (and thus it remains constant). So, the speed of a wave in water, for example, remains constant even if the source of the waves (say a paddle keeps moving). Frequency and wavelength may change but the speed remains constant. In a similar fashion, light is an electromagnetic wave and thus its speed is constant. And the constant speed is defined by the medium (which happens to be vaccum i.e., no medium for light).   My questions are twofold now -    Is the above reasoning correct to explain why speed of light is constant? (i.e., thinking of it as constancy of any wave in general) Theoretically, could there be other waves that could travel through vaccum at a different speed? If yes, time dilation and length contraction will have to be different.   I'd appreciate it if the answer is kept non-technical (essential math is okay). More importantly, I request you to try and explain without using General Theory of Relativity. Because I'm thinking how the scientific understanding by 1905 had already knew that speed of light has to be constant.
5487
I'm in a freshmen level physics class now, so I don't know much, but something I heard today intrigued me. My TA was talking about how at the research facility he worked at, they were able to accelerate some certain particle to "99.99% the speed of light". I said why not 100%, and I didn't quite understand his explanation, but he said it wasn't possible. This confused me. Since the speed of light is a finite number, why can we go so close to its speed but not quite?  Edit: I read all the answers, and I think I'm sort of understanding it. Another silly question though: If we are getting this particle to 99.99% the speed of light by giving it some sort of finite acceleration, and increasing it more and more, why cant we increase it just a little more? Sorry I know this is a silly question. I totally accept the fact we cant reach 100%, but I'm just trying to break it down. If we've gotten so close by giving it larger and larger acceleration every time, why cant we just supply it with more acceleration? And how much of a difference is there between 99.99% the speed of light, and the speed of light? (I'm not really sure if "difference" is a good word to use, but hopefully you get what I'm asking).