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The short answer to the first question is no. The point is to understand that light does not have to be given the special pedestal we give it while analysing problems such as above. Anything moving at a speed $c$, be it a photon or any other particle would follow the second postulate of relativity. On the other hand photons which don't travel at $c$ due to ...


11

Supposing a train gets accelerated Let's pause there for a moment. In Special Relativity, simultaneity is relative. That means that two event that appear simultaneous in one inertial reference frame are generally not simultaneous in another reference frame. This means that if two ends of a train began accelerating together in one inertial reference frame, ...


6

@IndischerPhysiker has already answered the question, but I would like to bring forward an example of Cherenkov radiation - a phenomenon that occurs when a particle moves through the medium faster than the speed of light (in the medium), and raises precisely the question voiced in the OP: Cherenkov radiation is electromagnetic radiation emitted when a ...


4

First, imagine a stopwatch at rest that measures a time interval $\Delta t$. This concretely means there are two events: event 1 is that we start the stop watch at time $t_1$ and position $x_1$, and event 2 is that we stop the watch at time $t_2=t_1+\Delta t$ while the watch is still at position $x_2$. The spacetime interval in the stopwatch frame is then \...


4

There is a simple intuitive picture for Lorentz length contraction and time dilation that can be understood by anyone familiar with ordinary geometry. It's not perfectly accurate, because spacetime is geometrically different from Euclidean space, but it's easiest to start with the Euclidean picture of what is going on, and then introduce the differences that ...


3

The relative speed of light in water was first determined by Fizeau who found experimentally that it depended on the motion of the medium. The combination of Fizeau's result and the null result of the Michelson-Morley experiment was the impetus for the development of special relativity. So the answer to your first question is that the speed of light in a ...


3

Does the location where the force is applied (the engine of the train) affects the contraction center? Information can travel at most with the speed of light. So if you apply force on the back of the train, the front will not know it should also accelerate until later. This means, the train would shrink in its own frame (i.e. really, physically shrink). ...


3

The infinite-dimensional, unitary reps are classified and constructed using the method of induced representations (basically the famous Wigner paper). The finite-dimensional, non-unitary or indecomposable, reps are not classified. To get a taste of what you’re getting into, check out this nice paper on the simpler E(2) case. I can’t find the reference but ...


2

A photon with energy $E$ propagating in the $+x$ direction in (for simplicity) 1+1D spacetime has four-momentum (well... two-momentum) $$p^\mu=\begin{bmatrix}E\\E\end{bmatrix}$$ where $c = 1.$ You can get this from 1) that $p^0 = E$ (the time component of any four-momentum is energy) 2) that $p^\mu p_\mu=0$ (the four-momentum is lightlike because photons are ...


2

The dot product on a Minkowski manifold is defined to have indefinite signature, so the second option is correct. P.S. uses $\eta_{\mu\nu} = \text{diag}(1, -1, -1, -1)$.


1

The scalar field transforms in the trivial representation of the Lorentz group, so $J^{\mu\nu} = 0$ for scalar fields. The equation $J^{\mu\nu} = \mathrm{i}\left(x^\mu\partial^\nu - x^\nu \partial^\mu\right)$ is only true when momentum is represented as $p^\mu = \partial^\mu$, i.e. this expression is for $J^{\mu\nu}$ acting on a wavefunction, not on a ...


1

It is some hypothetical material substance that permeates the universe. More precise definition depends on the particular theory of aether. With regard to electromagnetism its hypothetical material substance in which electromagnetic waves propagate in analogy to propagation of waves in water or sound in air. Because Maxwell theory is not invariant under ...


1

Relativistic momentum $p$ is the “spatial component of the particle 4-momentum”, whose magnitude reduces to $E/c$ for lightlike 4-momenta and $\gamma m v$ for timelike momenta.


1

Your question raises a common and interesting misconception. Time does not slow down from the perspective of a moving observer. The time dilation effect means that where a moving observer passes between two stationary clocks, the time experienced by the observer is less than the time difference recorded on the clocks. From the perspective of the moving ...


1

Your confusion stems from the fact that the second statement is wrong. We are not traveling with the speed of light -- in ANY frame of reference. Special relativity teaches us that an observer traveling at $c$ is traveling with this speed in any frame of reference. Therefore, if I see you standing next to me (and not traveling at c) then you are not so in ...


1

The easiest way, by far, to think about Penrose-Terrell rotation is in the rest frame of the object being looked at, not the rest frame of the rocket. In that frame, light from the object fills spacetime with a static light field. What you see at a spacetime point is determined only by your $x,y,z$ coordinates in that frame. Moreover, it's just what your ...


1

You seem to have a very fundamental misunderstanding of an observer in special relativity. In special relativity, an observer should not be thought of as an actual person located at one point in space. Rather, an observer is something that is everywhere and "instantaneously" knows all events at all points in spacetime. The word "observe" ...


1

You are missing a $t$, your commutator should be $$L_v\xi=\left[\frac{\partial}{\partial t}+s\frac{\partial}{\partial x^{1}},t\frac{\partial}{\partial x^{1}} \right]=\frac{\partial}{\partial x^1}-t\frac{\partial s}{\partial x^1}\frac{\partial}{\partial x^1}$$ Keeping in mind that $t=x^0$ and $s=\frac{x_1}{x_0}$, you have that $t\frac{\partial s}{\partial x^1}...


1

Since Feynman considers point particles let us discuss this case. Yes, OP's speculations can indeed be realized. The Hamiltonian action $$\begin{align}S_H[x,p,e]~=~&\int d\tau ~L_H, \cr L_H~=~&p_{\mu}\dot{x}^{\mu}-\frac{e}{2}(p_{\mu}p^{\mu}+m^2), \end{align} $$ for a relativistic point particle has the mass-shell constraint $$p_{\mu}p^{\mu}+m^2~\...


1

This is the form that is most useful for experiments. Suppose that we have a particle that emits some radiation with a characteristic frequency. If we send those particles through the lab at high speed then we can use the lab frame as the unprimed frame and the particle’s frame as the primed frame. We then use the known characteristic frequency in the ...


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