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Questions tagged [special-relativity]

The special theory of relativity describes the motion and dynamics of objects moving at significant fractions of the speed of light.

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Wigner-Eckard theorem in 3+1 Minkowski dimensions

From this source, I have: I cannot find much (if not any) information online for Wigner-Eckard in 4D, hyperboiloid harmonics etc. And there are many facts just states in this source that I would ...
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53 views

Converting an invariant matrix to a non-invariant tensor

I'm working on the following problem: In 4-dimensional notations, given a transformation matrix Calculate the matrices $\Lambda_{\mu\nu}$, $\Lambda_\mu^\nu$ and $\Lambda^{\mu\nu}$ The ...
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The Necessity of Light for the Synchronization of Clocks in Special Relativity

In Einstein's paper, "On the Electrodynamics of Moving Bodies" light is used to synchronize the clocks of two different points in space. Thus if a frame S' is moving relative to frame S, an observer ...
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52 views

Einstein's frozen wave?

Here's the quote from Stephen Hawking's latest book. At the age of sixteen, when he (Einstein) visualised riding on a beam of light, he realised that from this vantage light would appear as a ...
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How to write the Poincare transformation for an arbitary path in Minkowski space?

So lets Say for arguments sake we have some vector $V^{a}$ and we drag it along some path $\gamma_{1}$ in Minkowski space $R^{3,1}$. For a straight path (represented by a vector $\Delta\overrightarrow{...
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Error of relativistic kinetic energy

I have recently begun working on the special relativity theory. I have then made the taylor series for the gamma factor to show that we get the classic formula for kinetic energy: $$E _ { k i n } = m ...
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Poincaré and Galilei group - notation

On this slide it just says that $\mathcal{P}$ and $\mathcal{G}$ are the Poincoré and Galilei groups, but I do not understand what they are made of. What does $\mathbb{R}^{1,3}$ mean? Why does $\...
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How can zero point energy vacuum be Lorentz invariant?

What distribution of electromagnetic radiation is Lorentz invariant? How can radiation look the same regardless of inertial frame? According to Marshall and Boyer a cubic distribution like $ \rho(\...
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Direct Product vs Tensor Product

I am confused in the notation on page 67 and page 70 a text (http://www-pnp.physics.ox.ac.uk/~tseng/teaching/b2/b2-lectures-2018.pdf), whether it's talking about a direct product or an outer product: ...
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Lack of invariance of relativistic doppler effect for inertial frames?

The equation for the relativistic doppler effect in a medium where waves travel at speed $c_m$ is $$\frac{f_r}{f_s} = \frac{c_m - v_r}{c_m-v_s}\frac{\gamma_r}{\gamma_s}$$ My question is whether this ...
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If 2 spaceships approaches each other at 0.99c, how much blue shift would they see? [closed]

Imagine these 2 spaceships moving at 0.99c toward each other and speed isn't additive how much blue shift would they see? I know how to calculate when the scenario is spaceship vs Earth but how about ...
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Minimal Requirements for Space and Time

I read recently that to consider the Planck length of less than 1.6 x 10 exp -34 meters the smallest unit that can manifest 4 coordinate space time to be incorrect in its interpretation. The reason ...
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Having trouble making sense of Einstein's thought experiment

So I was reading about Einstein's thought experiment where he tries to show that simultaneous events in one frame may not be simultaneous in another frame. So, in the given pic, light from B' ...
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44 views

Four-velocity and the metric tensor

The metric tensor $g_{{\mu}{\nu}}$ has this property $$g_{{\mu}{\nu}}g^{{\mu}{\nu}}=4$$ and the four-velocity, $U^{\mu}=\frac{dx^{\mu}}{d\tau}$ which has this property $$U_{\mu}U^{\mu}=c^2.$$ So ...
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Do vibrations increase an object's mass or weight?

does a vibrating object recieve an increase of mass or weight? and if it does, at what frequency or intensity does it need to vibrate at, and what is the rate of increase? is there a formula for it? ...
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107 views

Simultaneous measurements in Quantum Mechanics

In the notes I am using, it states that if two observables A and B are measured simultaneously, then the measurement of A does not affect the measurement of B, and vice versa. However, why does the ...
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Coordinate transformation for a comoving frame for abitrary trajectory

Let's suppose there is a particle moving through space time. It moves along the x-direction with a time dependent velocity. Seen from an inertial frame it has the velocity $v(t)$ where $t$ is the ...
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beta velocity notation in Galilean matrix

I was looking at the Galilean transformation matrix and came across a matrix for Galilean of the form. $$\begin{pmatrix}x'\\ y'\\ z'\\ t'\end{pmatrix}=\begin{pmatrix}1&0&0&-\beta c\\ 0&...
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Doubt in Weinberg's book on Quantum Field Theory

In page number 59 of his book on QFT, Weinberg mentions that for the operator $U$, defined for infinitesimal parameters $\omega$ and $\epsilon$ as: \begin{equation} U(1+\omega,\epsilon)=1+\dfrac{1}{2}...
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Why is it important that the combination of charge, parity & time reversal symmetry not violated?

If looking for more particles or decays that violate CP symmetry can explain why there is so few antimatter in the known universe, I guess finding things that violate CPT symmetry might helps clear up ...
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Converting a general QFT state to another inertial of light-cone reference frame

Assume, in a certain reference frame, a relativistic QFT at time $t=0$ is in the state $$ \hat{\Psi}(t=0) |vac\rangle \quad, $$ where $$ \hat{\Psi}(t) = \operatorname{e}^{-i\hat{H}t} \sum \limits_{n=1}...
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Angular momentum of a circualr polarized EM wave

In an exercise, I am being asked to compute the angular momentum of a circularly polarized wave. The wave is defined by the four potential: $$\Phi^\mu(x^\nu) = \text{Re} \left\{ \varepsilon^\mu e^{...
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Relativistic Mass when Viewed from Different Frames of Reference

I understand relativistic mass and the equations underpinning it. My question deals with how to calculate relativistic mass when an object is viewed from different frames of reference. Consider a ...
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Can the Dirac Hamiltonian accommodate a variable speed of light?

The Dirac Hamiltonian has the form1 $$\left[\beta m c^2+c\sum_{n=1}^3\alpha_np_n\right]$$ where $\alpha_n$ and $\beta$ are Hermitian matrices, and $c$ is the speed of light. My question: Is there a ...
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Another objection to Feynman's moving infinite sheet of charge “radiator”

In The Feynman Lectures on Physics Vol II 18-4, A travelling field we are provided the thought experiment of two uniformly and oppositely electrically charged planar sheets of infinite extent lying in ...
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How are these two definitions of energy related/derive from each other? [closed]

How are these two definitions of energy related/derive from each other: $$E = \frac{mc^2} {\sqrt{(1 - v^2/c^2)}}$$ vs. $$E = \sqrt{m^2c^4 + p^2c^2}~?$$ Both equations express the relation between ...
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Consider two electrons released from rest at a distance $d$. How much energy is radiated?

Abraham-Lorentz? Two electrons d apart have potential energy. Release them, they will be repelled according to Coulomb's law. I could make an assumption about the associated vector potential, but I ...
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1answer
34 views

Can the Hermitian conjugate of a column vector still be column vector?

From what I understand, contravariant vectors are represented by column vectors, and covariant vectors are row vectors. So for a QED current, say $j ^ { \mu } = \overline { \psi } \gamma ^ { \mu } \...
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$(1,1)$ representation of $SL(2,\mathbb{C})$

How do you prove that the $(1,1)$ representation of the $SL(2,\mathbb{C})$ group acts on symmetric, traceless tensors of rank 2?
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Showing the Lorentz-Fitzgerald length contraction

I'm trying to show that if we have a rod of length $L$ that is at rest wrt the inertial frame $K$, and we have a observer $K'$ moving wrt $K$ with speed $v_0$ in the direction along the length of the ...
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Does the inertia of an object changes with its temperature? [closed]

$F=ma$ is not accurate enough at relativistic speeds. Does the inertia of a rigid body related to its temperature?
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A Non-Velocity Dependent 4-Vector Transformed as Velocity Dependent 4-Vector

I would appreciate it if someone could explain the following: It is understandable that the Minkowski space Gauss' and Ampère-Maxwell equations in index notations \begin{equation} \partial_{\nu} F^{\...
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1answer
134 views

What if a spaceship going half the speed of light turned on its headlights in both directions?

If the spaceship, travelling at c/2 were to turn on only its forward facing headlights, then after one second (relative to us, the stationary external observers) we would see that the ship has moved ...
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information theory and energy mass relationship (E=mc2) - just a hunch … [closed]

I was walking on the street the other day and thought to myself, maybe E=mc2 can be derived from information theoretic principles. I did not do too much work on this, just want to brainstorm a bit. ...
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Do lengths contract or expand due to relativity?

Ok I have a simple question. We consider two frames, one is $F$ which is at rest and one is $F'$ which is moving with $v_0$ in the $+X$ direction with respect to $F$. Suppose a rod is at rest in $F$ ...
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Space time diagrams : Length contraction

I am trying to get a better understanding for length contraction in a geometric sense and came across this diagram: but what I am having trouble with is why in the ref frame of the blue vector ...
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What happens to a uniform electric field when a point charge is sent moving in a direction neither parallel nor perpendicular to the field?

If you have two reference frames, one where a charged particle is stationary but in a uniform electric field, and another where the system is moving at speed $\vec v$ in a general direction (neither ...
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1answer
103 views

Surjectivity of momentum mapping

I have to show that the following mapping of momenta is surjective. The mapping $\{p_i^{\mu},p_j^{\mu},p_k^{\mu}\}\rightarrow\{\tilde{p}_{ij}^{\mu},\tilde{p}_k^{\mu}\}$ is given by $$ \tilde{p}_k^{\...
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3-particle phase space in $d$ dimensions

recently I came across a problem concerning the 3-particle phase space. I am trying to show, that the 3-particle phase space for massless particles with momenta $p_1$, $p_2$ and $k$ is given by $$ d\...
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A bus and a ditch - an extension of the barn ladder paradox [duplicate]

Suppose, a bus is travelling with constant velocity in a straight road, and there is a ditch in front of it. Suppose, the rest length of the bus and the ditch are equal. Suppose the speed of the bus ...
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2answers
86 views

How do we know that light cannot travel faster than it does?

We assume the speed of light in vacuum is its maximum speed but can we not assume that it could be faster, or slower?
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1answer
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Lorentz transformation of intensity

I have come up with an expression for the way intensity transforms, however it is different to the book, and I cannot seee where I have gone wrong. So we both start the same way: $$ \frac{I}{I_0}=\...
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1answer
73 views

Is speed of light is the only way to measure time [closed]

After reading up on special relativity, I understood that because of Einstein's postulates -the speed of light in free space is a universal constant Everytime we measure time for an event, directly ...
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1answer
100 views

Special relativity: I arrive at a contradiction regarding the Lorentz invariance of certain quantity

I want to show the Lorentz invariance of $d^3 p/E$ (Eq. 8.11 of Mandl-Shaw), where $E$ is the relativistic energy. Peskin-Schroeder gives sort-of, a proof in section 2.3 which I am convinced of. But ...
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Relation of the Lorentz group to $O(1,3)$

Let $\Lambda$ be an element of the Lorentz group. It satisfies the identity:$${\Lambda}^T\eta\Lambda=\eta$$ where $\eta$ is the Minkowskii metric. Hence by the usual definition of orthogonality, $\...
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Can spherical harmonics be used in relativity equations?

I have a neutral pion of mass $m_{\pi}$, and it decays into two photons. In it's reference frame the decay is isotropic. One of the photons has a helicity of $+\hbar$ and the other $-\hbar$. In ...
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What are the relativistic arguments for existence of space? [duplicate]

One of the interpretation of special relativity states that the time is an illusion and does not really exist. This interpretation promoted Einstein himself in his condolence letter after the death of ...
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Why am I not getting the right values for the time it takes a spaceship to get to 0.9c? [duplicate]

I'm attempting this problem: Rocket undergoes constant acceleration from rest until it reaches a speed of 0.9c. The astronaut in the rocket feels a constant "induced gravity" of 1g. How long does it ...
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2answers
62 views

Question about derivation of four-velocity vector

In order to describe a notion of rate of change of positon, in four-dimensional spacetime, we have to introduce the concept of four-velocity. So, consider the following: For a massive particle ...
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1answer
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Relativistic Doppler Effect - calculate frequency $f'$ and $f$ in references $S'$ and $S$

Considering that $S$ (fixed) and $S'$ are inertial references which coincides when $t=t'=0$ and their coordinates $(ct,x,y,z)$ and $(ct',x',y',z')$ are related by Lorentz's transformations, calculate ...