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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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 ...
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36 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 $v$...
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32 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. ...
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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 ...
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47 views

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}^...
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64 views

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}}{...
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67 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 \\...
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Mother wavelet and Lorentz invariance

Can we choose a mother wavelet that is a Lorentz variant in practical applications of wavelet transform in Minkowski space?
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221 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 ...
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1answer
31 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 $t$? ...
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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}{...
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177 views

What is the conceptual idea behind raising and lowering indices?

I've been watching Eigenchris' playlists on Tensors for beginners and Tensor calculus. His videos really clear up a lot of concepts. In the last video of the Tensor for beginners series, he talks ...
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50 views

The causal cones are convex

Taking into account the definitions of causal, temporal, spatial and luminous vectors, I want to test the following results: Each temporal cone is convex. The proof of this first result would be the ...
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134 views

Inscribed angle theorem for hyperbola - is this result well known?

I (re-)discovered the following result recently. Inscribed angle theorem for hyperbola The Minkowski inner product on $\mathbb{R}^{1,1}$, $u\cdot v=u_1v_1-u_2v_2$, satisfies a version of the Cauchy-...
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Calculating the "boosts" of $(M,g)$

Consider a Lorentzian manifold $(M,g)$ equipped with its metric: $$ g=ds^2=\frac{dudv}{uv}-\frac{dr^2}{r^2}-\frac{dw^2}{w^2}. $$ How do you calculate the "boosts" of $(M,g)?$ I need to ...
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62 views

Einstein says that this equation is true, but WolframAlpha says it's not always true. Who is right?

Einstein says $$\cos\mathrm{i}x=\frac1{\sqrt{1-\left(\mathrm{i}\tan\mathrm{i}x\right)^2}},$$ but WolframAlpha says that this isn't true for $x=\pm2$ and $x=\pm9/5$. What's happening? From page 34 of ...
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Making sense of Wikipedia derivation of the invariance of the spacetime interval

In this page there is a reproduction of an argument (I believe originally due to Landau?) that the interval between two events in spacetime is independent of the observer. I'm trying to make more ...
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1answer
298 views

Lagrangian equation compute conserved angular and linear momentum.

Question Consider the following Lagrangian for a particle of mass $m$ ($c$ is a constant) $L = −mc \sqrt{c^2 - r' .r'} $ Show that it is invariant under translations, rotations and compute the ...
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59 views

3-dimensional slice of $\zeta^{2,2}$

Consider a semi-Riemannian manifold $\zeta^{2,2}$ with metric, $g=\frac{dxdy}{xy}+\frac{dudv}{v-uv}.$ How could you define a 3-dimensional slice of $\zeta^{2,2}$? What would it look like? I guess ...
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486 views

Killing vectors in Minkowski Metric

Firstly, I know this is a physics-related problem, and I have posted here, but the physics forum seems so much more empty then this one, so here it goes: I was in the process to find the Killing ...
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When can you divide differential 1-forms?

I am reviewing some general relativity, starting specifically with flat space (i.e. special rel) and came across a derivation of the velocity transformation between reference frames that I had an ...
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Spinors and Klein-Gordon Equation

I'm currently working through Chapter 13 of Wald's General Relativity and spinors are being a little illusive to me. The question is pretty much: Using the Klein-Gordon equation in the form: $$\...
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1answer
50 views

Domain issues in transformation of the coordinate representation of a function

Start with a Manifold $M$ and define a function $f:M\rightarrow\mathbb{R}$. As usual, pick two charts $(U,x)$ and $(V,y)$ with $p \in U\cap V$ and $x:M \supset U \rightarrow x(U) \subset\mathbb{R}^n$. ...
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1answer
21 views

How to propagate the uncertainty of momentum through the relativistic velocity equation?

I have the standard deviation of momentum as $σ_p$ and I am trying to find $σ_v$. The equation I want to propogate uncertaintiy through is the relativistic velocity equation: $v=\sqrt{(p/m)^2/(1+(p/m)^...
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1answer
32 views

Deriving $KE=0.5mv^2$ from the relativity equation

From this article: http://hyperphysics.phy-astr.gsu.edu/hbase/Relativ/releng.html#c6 It shows how to derive the classical formula for Kinetic energy from Relativistic Kinetic energy. I've started at ...
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88 views

How does calculus work on a Pseudo Riemannian manifold?

I'll first recapitulate the why calculus works for Riemannian manifolds, and then present why I believe the same does not work for Pseudo-Riemannian manifolds. I'd like to know where my current model ...
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1answer
93 views

How does topology work when taking charts on a Psuedo-Riemannian manifold?

I'll first explain why I think taking charts is sane when working with Riemannian manifolds, and then show what I believe breaks down in the Pseudo-Riemannian case with a particular choice of a Pseudo ...
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2answers
141 views

Help with proof involving the Minkowski metric tensor.

$\newcommand{\LF}[2]{\Lambda^#1_{\hspace{.2cm} #2}} \newcommand{\LL}[2]{\Lambda^{\hspace{.2cm} #2}_#1} \newcommand{\af}{\alpha} \newcommand{\be}{\beta}$ I'm trying to prove that the Minkowski metric ...
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1answer
73 views

What is the Minkowski Metric?

I recently asked "What is the Metric Tensor?" and a very helpful answer from @R.N.Raia gave me a much better understanding as to what it is. The only problem is that there are a few terms their ...
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140 views

Deriving the relativistic rocket equations.

I am trying to derive the relativistic rocket equations found here [(4),(5),(6),(7),(8)] but I do not understand proper time, proper velocity and proper acceleration. Define a point $P$ with ...
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35 views

Special relativity confusion [closed]

Alex is on Earth and the planet Flez is 5 light years away. Glen, is on a spaceship approaching Earth at $0.8$c along a direct path towards Flez. Unfortunately Flez explodes. According to Alex, this ...
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143 views

Why the 4-acceleration in MCRF is not $(0,0,0,0)$?

As you all know, the 4-velocity vector components in the momentarily comoving reference frame (MCRF) is given by $$\vec U=(1,0,0,0)\ .$$ On the other hand, the 4-acceleration vector is given by $$\...
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37 views

How does one derive the Lorentz transform for rectangular hyperbolas?

I've only seen derivations of the Lorentz transformation using hyperbola like $x^2-y^2=-1$ as opposed to hyperbola like $y=1/x.$ I've been trying to derive the Lorentz transform for rectangular ...
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1answer
43 views

Relativistic Limit [closed]

Excuse me. I have a problem about double limit, namely, Relativistic Limit. Let $c$ is positive real constant. Calculate \begin{equation} \lim_{v\to c}\lim_{V\to c}\frac{v - V}{1 - vV/c^2}. \end{...
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1answer
58 views

How can we find the Lorentz Factor using these equations?

$$ct'=\gamma(ct-vt)$$ $$t'=\gamma\Bigg(t-\dfrac{\gamma^2-1}{\gamma^2v}x\Bigg)$$ I'm trying to understand how did the textbook derivate the Lorentz Factor just using these two equations.
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70 views

The intersection of $w = x / \beta$ and a line parallel to $w = \beta x$.

If you are familiar with physics, you may have guessed what the background is… but, the following is simply a question of elementary geometry. (Not consider the special relativity.) Question: ...
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56 views

Why does this method of using trigonometric functions to calculate relativistic gamma work?

When I studied special relativity, I noticed that some of the problems and answers on calculating $\gamma$ would be fractions of c that looked like ratios of side lengths in right triangles. For ...
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Parametrization of the Lorentz Group

The Lorentz group is the group of $4\times 4$ matrices that satisfy $$\Lambda^T \eta \Lambda = \eta, \eta = \begin{bmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 ...
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20 views

Relativistic velocity transformation inequality

Suppose $-c<u<c$ and $-c<v<c$. Prove that $\displaystyle -c<\frac{u+v}{1+uv/c^2}<c$. My attempt: Adding the two given inequalities, we get $-2c<u+v<2c$. $|u|<c$ and $|v|&...
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49 views

Show that a reference frame exists where the spatial separation is zero.

question I am trying to show that if the Lorentz Invariant Interval is positive $$c^2 \Delta t^2 - \Delta x^2>0$$ then there exists a reference frame $S^{'}$ where $\Delta x^{'}=0$. context ...
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1answer
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$A^t\eta A=\lambda\eta $ then $\lambda$ is positive

Let $A$ be any real $4$ by $4$ invertible matrix such that $A^t\eta A=\lambda\eta$. Here $\eta=diag(-1,1,1,1)$. Then I have to show that $\lambda$ is positve. I tried the determinant but it gives $\...
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1answer
99 views

$SL(2,\mathbb{R})$ as a Lorentz Group $O(1,2)$

Define $$X = \bigg\{ \begin{pmatrix} x_0+x_1 & x_2 \\ x_2 & x_0-x_1 \end{pmatrix} | x_i \in \mathbb{R} \bigg\}.$$ Given $g\in SL(2,\mathbb{R})$, consider $s(g)$ which is a transformation $x \...
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1answer
38 views

Derivative of (Norm of) Möbius Addition w.r.t. Curvature

Does anyone know a resource showing the formula for the derivative of the (norm) of the Möbius addition? This is the Möbius addition: $$ x \oplus_cy=\frac{\overbrace{(1+2c\langle x,y\rangle+c||y||_2^...
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1answer
106 views

Do null geodesics have zero acceleration?

In V. Faraoni's textbook "Special relativity", on p. 173 we find the statement "A parameter $\lambda$ such that $$\frac{d^2x^\mu}{d\lambda^2}=0$$ is called an affine parameter." (Here $x^\mu=x^\mu(\...
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102 views

How to understand Non-positive Definite Metric Tensors intuitively?

My motivation for this comes from Minkowski space and general relativity more broadly, but I don't want to focus on the real world details currently because I want to understand what a positive vs ...
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18 views

How to check satisfiability of a large number of "lorenzian" quadratic inequalities

Given a list of $m$ vectors $x^i=(x^i_t,\textbf{x}^i)\in\mathbb{R}^{n+1}$, $i\in \mathbb{Z}_m$ and two disjoint sets of vector pairs $A,B\subset \mathbb{Z}_m\times\mathbb{Z}_m$ as well as a set $C\...
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1answer
29 views

Index notation problem in getting $(\Lambda^{-1})^{\mu}_{\nu}=g^{\mu\alpha}g_{\nu\beta}\Lambda^{\beta}_{\alpha}$

Let consider: $g_{\mu\nu}=diag(1,-1,-1,-1)$ and $\Lambda$ so that $\Lambda^{T}g$$\Lambda$=g I want to prove that $g=g\Lambda\Lambda \Rightarrow \Lambda^{-1}=g^{-1}g\Lambda$ The proof is silly for ...
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40 views

Second countability of Minkowski double cones.

In the 4-dimensional Minkowski spacetime, for a given point $x=(x^0,x^1,x^2,x^3)$, its timelike future or past set is defined as, $I^{\pm}(x)= \{y=(y^0,y^1,y^2,y^3) \in \mathbb{R}^4: \eta_{\mu \nu}(y−...
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1answer
1k views

Prove derivative of contravariant tensor of rank 1 is a mixed tensor of rank 2

$A^\alpha$ is a given contravariant vector (when $\alpha\in {0,1,2,3}$ a $4$-vector in Minkowski space) I need to show that the derivative $\frac{\partial A^\alpha}{\partial x^\beta}$ is a mixed ...
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65 views

Is there a closed form for the recurrence $V_{n+1}={V_n+\Delta V\over 1+{V_n\cdot \Delta V/C^2}}$, for constants $\Delta V$ and $C$?

I was wondering if the following recurrence formula has a closed form: $$V_{n+1}={V_n+\Delta V\over 1+{V_n\cdot \Delta V\over C^2}}$$ where $\Delta V$ and $C$ are positive constants, $V_n$ is the ...