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Questions tagged [special-relativity]

For questions relating to Einsteins special relativity theory, the equivalence of physical laws in different inertial frames.

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Does time always dilate? [migrated]

Consider an immobilised system O and a moving with speed v relative to O system O’. By (x,t) we denote the spacetime coordinates of a point. For two events A,B happening in O consider the following ...
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Need some Help about understanding classifications of projective representations

I have trouble with understanding the correspondence between projective representations of a connceted Lie group and linear representations of another Lie group. This is the Bargmann's theorem: Let $G$...
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Two dimensional Lorentz group

The Lorentz group $O(D-1,1)$ in $D$ dimensions is the group of linear transformations $\mathbb{R}^D \to \mathbb{R}^D$ that preserves the Minkowski metric $(\cdot ,\cdot ):\mathbb{R}^D\times \mathbb{R}^...
Mahtab's user avatar
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Relativistic Phase Space for Three-Particles

I am studying introductory quantum field theory and I am trying to reduce the integral over the relativistic phase space of three particles of arbitrary masses to an integral over two energies and ...
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The group $SO(1,3)$ and its Lie algebra

Denote the matrix $\eta=$ diag$(-1,1,1,1)$. The group $O(1,3)$, called Lorentz group, is the group of all matrix $L\in M_4(\mathbb R)$ such that \begin{align} L^\top\eta\,L\,=\,\eta.\tag1 \end{align} ...
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How to derive Lorentz (general) transformation in Special Relativity?

Let $M$ be a 4-vector space over $\mathbb R$. Since $M$ is linearly isomorphic to $\mathbb R^4$, the linear isomorphism is also a diffeomorphism between $M$ and $\mathbb R^4$. For each $p\in M$, we ...
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SRT - Distance to the moon for a rocket

I shall use units where time and distance are in seconds and speed is dimensionless between zero and one. Suppose a rocket $B$ passes by the earth $A$ on its way to the moon at speed $v$. At that ...
alexanderyaacov's user avatar
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Mathematical theory of plasma

I am working on a heavily mathematically project about plasma. In particular, I want to find references that treat the problem from microscopic models that include relativistic and magnetic effects (...
The N's user avatar
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An unzipping problem

Imagine a continuous one-dimensional line, which is duplicated exactly once. Duplication starts at random spacetime points. Once a point is duplicated, it starts a double duplication wave moving in ...
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How Would a Rapidly Spinning Unit Sphere Appear to an External Observer Due to Relativistic Effects?

I am trying to visualize and calculate how a unit sphere would appear to an observer if it was spinning so fast that the tangential speed at the equator approached the speed of light. Here is the ...
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How to differentiate a light-cone integral in relativity?

Consider the following integral $$ \int_{V_X(v)}I(Y)\,dY\tag1 $$ In light cone coordinates $x_\pm=x\mp vt$, where $t$ is time and $v\in\mathbb{R}^+$, $V_X(v)$ is the past light cone, given, by $$ V_X(...
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Components of a vector in Hyperbolic Geometry

I was recently studying "The classical theory of field' by L.Landau. In the introduction the book introduced theory of relativity. (I know I should be asking such questions on physics stack ...
Charu _Bamble's user avatar
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Tensor Index Manipulation

I am trying to study General Relativity and I thought about starting with some index gymnastics. I found a worksheet online and I am stuck with a simple problem. I have to prove that $\partial_{\mu} g^...
 Paranoid's user avatar
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Hyperbolic and Conformal Geometries applied to Relativity Physics - Are there other geometries?

If we analyse the light spheres in two inertial frames $K_1$ and $K_2$ we have $$x_1^2+y_1^2+z_1^2-c_1^2t_1^2=0$$ $$x_2^2+y_2^2+z_2^2-c_2^2t_2^2=0$$ (making no assumptions as the constancy of $c$ at ...
James Arathoon's user avatar
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1 answer
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How to find aberration angle when observing a distant star via a telescope from earth?

I am reading a book called "Special Relativity" by Anthony Philip French. There is a small section that discusses the situation of trying to observe a distant star using a telescope from ...
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How would the front and back of the big bang appear in a 3-sphere universe?

I'm asking purely from a mathematic point of view - How would the front and back of the big bang appear in a 3-sphere universe? In particular, if the big bang is thought of as a 2-sphere and if we ...
it's a hire car baby's user avatar
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How many objects can I perceive to all be travelling at speed arbitrarily close to $\sqrt 2\cdot c$ relative to each other?

Classically, space is a $\Bbb R^3$ manifold. According to relativity, spacetime is $4$d although I understand this is Minkowski space. The thing I was curious about, is whether the two theories ...
it's a hire car baby's user avatar
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Two spacecraft leaving earth-Doppler relativistic effect

I recently begun approaching special and general relativity. I found this problem in my book and I'm trying to solve it but I'm finding some difficulties: An observer on Earth sees two twins A and B, ...
Adam's user avatar
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Lorentz transformations of energy and relativistic 3-momentum

My question is how do energy and the relativistic 3-momentum change under a Lorentz transformation? I know that the 4-momentum transforms as $P' = \Lambda P$ where $\Lambda$ is a $4\times 4$ Lorentz ...
idk31909310's user avatar
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How can I derive the First Bianchi Identity of the Riemann Tensor from this particular consideration.

I have been asked to consider the antisymmeterisation: \begin{equation} \nabla_{[\mu}\nabla_{\nu}\omega_{\lambda]}. \end{equation} By expanding out this expression, I have found: \begin{equation} \...
the fart king's user avatar
1 vote
1 answer
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"Double" relativistic variant of the same classical mechanics equation

This question is about my curiosity about the relativistic Kepler equation of which I am reading in a recent paper. Actually, I am only interested in an introductory concept stated in paper. Let $$ m\...
C. Bishop's user avatar
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Examples of relativistic equations

I am posting this question here because it is just reference request and I do not need a fully detailed answer. Attending my physics class, we introduced two relativistic equations: $$ \frac{d}{dt}\...
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Isomorphisms of causal structure of space with Lorentz form

Let $E$ be a real vector space, and $q$ be a quadratic form on $E$ of signature $(-1,1,\cdots,1)$. Let us call time vectors the vectors $v$ such that $q(v) < 0$. The set of time vectors has two ...
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Manifolds where $\Delta t$ is conserved for any trajectory.

If we consider a medium where perturbations always take the same time to reach every point in space when measured from an arbitrary observer's frame of reference then how would length have to contract ...
manoroli's user avatar
4 votes
1 answer
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Using 4 by 4 Lorentz boost matrices to verify the tensor transformation law, $T^{\mu'\nu'}=\Lambda^{\mu'}_\alpha\Lambda^{\nu'}_\beta T^{\alpha\beta}$

In the following question, $K^{\prime}$ is a frame moving in the positive $x$-direction with speed $v$ relative to frame $K$: A tensor of type $(2,0)$ is $16$ numbers, $T^{\mu\nu}$ with the ...
Electra's user avatar
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Derive the Lemma of Christoffel symbol

Derive the Lemma of Christoffel symbol, $$\frac{\partial^2x^\rho}{\partial\bar x^\mu\partial\bar x^\nu}=\Gamma_{\mu\nu}^\gamma\frac{\partial x^\rho}{\partial\bar x^\gamma}-\frac{\partial x^\alpha}{\...
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Partial derivatives of covariant and contravariant four-vectors

Consider a four-vector, with all transformations $\Lambda$ being Lorentz transformations. Under a Lorentz transformation, the original four-vector goes to its transformed version (tilde) according to: ...
MrStealYourFrog's user avatar
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Minkowski Honeycombs

What are the honeycombs (tessellations) of Minkowski space? Would like to know at least the isochoric/cell-transitive/"space filling" ones, but a complete list of regular ones would be nice. ...
Eriek's user avatar
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Using hyperbolic geometry to derive Lorentz transforms

I want to know how to derive the below Lorentz transformation formula using hyperbolic geometry $$ct' = ct \cosh\phi - x \sinh\phi$$ $$x' = x \cosh\phi - ct \sinh\phi$$ Here is a spacetime diagram ...
Ray Siplao's user avatar
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1 answer
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How do you isolate and compute the corresponding velocity when inputting a specific magnitude of length contraction?

How do you isolate the velocity term from within the square root - $lo\sqrt{1-\frac{v^2}{c^2}}$ so that if I'm given the amount that length has been contracted, I can then calculate the corresponding ...
Python House's user avatar
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1 answer
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How can $\frac{\partial \phi}{\partial t}$ be expressed as a function of $\frac{\partial \phi}{\partial \tau}$ given relativistic coordinates?

If $\phi(x,y,z,t)$ is a scalar field, given the relation between Minkowski Space coordinates $c^2(d \tau)^2 = c^2(d t)^2 - (d x)^2 -(d y)^2 -(d z)^2$, how can $\frac{\partial \phi}{\partial t}$ be ...
Arkaevis's user avatar
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2 answers
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Given an equation of differentials, how can I find the relative rate of change

Context I am studying special relativity. In [1], Gray derives an equation that is written in terms of three differentials. Namely, $$\left(d\tau\right)^2 = \left(dt\right)^2 - \frac{\left|d\mathbf{r}\...
Michael Levy's user avatar
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5 votes
1 answer
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How is the Lorentz group, $\text{O}(1,3)$, defined using set theoretic notation?

Context I am studying special relativity. I am trying to understand how to define the group elements of the Lorentz group, $\text{O}(1,3)$. I understand from [1] that the Lorentz group is has (at ...
Michael Levy's user avatar
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2 answers
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How to prove that all Minkowski spacetime isometries (transformations in Poincare group) are compositions of translation and Lorentz Transformations?

It is said in wikipedia that Minkowski spacetime isometries, i.e. the transformation that preserves $$ (x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2-(t_1-t_2)^2 $$ between points, can be represented as $\mathbb{...
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World line and light cones in two dimensions

The following is an example of hartle : Consider the two dimensional metric ds^2 = -X^2 dT^2 + dX^2 And the world line X(T)=A cosh(T) where A is a constant with the dimensions of length. The light ...
Talha Ahmed's user avatar
3 votes
3 answers
616 views

Can a manifold be reconstructed from its charts?

I'm learning special relativity and I am having a confusion on this mathematical point. Whenever any sort of motion or non motion happens in the world, it can only be perceived by the scientist in a ...
Babu's user avatar
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How the green function for the relativistic heat equation converges to the green function of the heat equation?

The relativistic heat equation or telegraphers equation is: $$ (\alpha\partial_t^2 + \beta\partial_t - \omega\,\nabla^2_{\text{3D}})G_R = \delta $$ if $\alpha \rightarrow 0$ the solution must ...
Alessandro Bossi's user avatar
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Physical theories and Mathematics

I study pure mathematics. In pure mathematics, we begin from some axioms and obtain theorems. I am also interested in studying physics. I have two questions about the relationship between physical ...
S Ali Mousavi's user avatar
1 vote
0 answers
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an apparent "inconsistency" in lorentz transformations?

why the Lorentz transformations forbid passing to a reference system with the speed equal to that of light despite there are particles (photons) traveling at this speed? in other words I see the fact ...
user273366's user avatar
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2 answers
745 views

Einstein notation superscript vs subscript?

Hi I'm new to Einstein notation when describing position in 4 dimensions. I understand that $x^μ=(x^0,x^1,x^2,x^3)$ represents $t, x, y$ and $z$ but I'm having a hard time understanding what $x_μ$ ...
Joel100's user avatar
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1 vote
0 answers
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Transformation of line element in Minkowski space under infinitesimal coordinate variation

I'm trying to understand the following problem: We are looking at an infinitesimal coordinate transformation $$ x^\mu \rightarrow x^\mu + \epsilon u^\mu(x), \space \epsilon \rightarrow 0 $$ and we are ...
diesmond's user avatar
1 vote
1 answer
84 views

Taylor Polynomial in Physics-related Question

So I've got a function here: $$m(v)= \frac {M}{\sqrt{1- \frac{v^2}{c^2}}}$$ which basically states that the mass $m$ of an object with rest mass $M$ (a positive constant) changes with its velocity $v$...
Jacques C.'s user avatar
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0 answers
41 views

Perturbation Theory Ambiguity?

I am trying to solve problem 2.1 in Schwartz, which is to derive the transformations $x \rightarrow \frac{x+vt}{\sqrt{1-v^2}}$ and $t \rightarrow \frac{t+vx}{\sqrt{1-v^2}}$ in perturbation theory. ...
Joeseph123's user avatar
1 vote
0 answers
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Calculate the initial energy in terms of M and m, using special relativity.

I have this problem and i don't know how to continue... A radioactive nucleus $A$ of mass $M$ moves forward with energy total $E_A$. It decays in flight to its stable state of mass $m$, emitting a ...
DeiYam's user avatar
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1 answer
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Wave equation under Galilean transformation

In Jackson's book on classical electrodynamics (3rd ed, ch 11, p. 516), he mentions how a wave equation for a field $\psi(\bf{x}^{'},t^{'})$ is transformed under Galilean shift, defined as $\mathbf{x}^...
user135626's user avatar
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1 answer
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How do you prove these definitions of $\cosh$ are equivalent?

I am reading the book, Geometry of Special Relativity, by Tevian Gray. In the introductory chapter to hyperbolic geometry, he states that the definition: $$ \cosh(\beta) = \frac{e^\beta + e^{-\beta}}{...
Luke Johnston's user avatar
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1 answer
127 views

Solving the Compton Scattering

This question refers to the Compton Scattering. We have an elastic impact between a photon and an electron, so conservation of $E$ and $\vec{p}$ in a 2D plane: $$\begin{cases}E^i_p+E^i_e=E^f_p+E^f_e \\...
Noumeno's user avatar
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3 votes
0 answers
306 views

Rigorous proof of time dilation (using only 1 spatial dimension).

$\newcommand{\set}[1]{\{#1\}}$ $\newcommand{\mc}{\mathcal}$ $\newcommand{\R}{\mathbf R}$ $\newcommand{\ST}{\mathbf S}$ Introduction The purpose of this post is to understand the theorem of time ...
caffeinemachine's user avatar
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1 answer
56 views

What is the hyperbolic angle as a function of $f(t)$ and, in general, two points?

What is the hyperbolic angle as a function of $f(t)$ and, in general, two points $(f(t),t)$ and $(f(t+a)$,$t+a)$? Is the following a valid way to define hyperbolic angle using $f(t)$ and $t$? ...
ajay's user avatar
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0 answers
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Intuitive way of seeing velocities in Zero Momentum Frames

I have two particles, one of them is stationary, another one was speed $u$ in the lab's frame. I cna prove with Lorentz transformations that the speed of the zero momentum frame is $$v=\frac{\gamma_u}{...
zabop's user avatar
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