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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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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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Mathematical trivial double about lorenz gauge $\partial_{\mu}A^{\mu}$

$$D_{\mu}.=\partial_{\mu}+iqA_{\mu}$$ $(D_{\mu}D^{\mu}+m^2)\phi(x)=0$ Expliciting: $(\partial^{\mu}\partial_{\mu}+iq\partial^{\mu}A_{\mu}+iqA^{\mu}\partial_{\mu}-q^2A^2+m^2)\phi(x)=0$ Setting Lorenz ...
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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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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 ...
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Calculation of Derivatives of Tensors in Index Form

If I wanted to show that the energy-momentum tensor $T^{\mu\nu}$ is not uniquely defined, I could add a term $\partial_{\lambda}\partial_{\lambda}X^{\lambda\mu\nu}$ to it where $X^{\lambda\mu\nu}$ = $-...
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Why the characters of the Minkowski spacetime translation group involve the Minkowski metric?

The translation group of Minkowski spacetime is just the additive group $\mathbb{R}^4$. Indeed, if $x\in \mathbb{R}^4$ is a point in Minkowski spacetime, the translation $T_v$ acts on $x$ by $$T_vx=x+...
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I wanted to know of book suggestions that can help me overcome my fear of indices

I want to go deeper into General Relativity and Tensor Analysis. However, manipulating the indices always seems to overwhelm me. I wanted to know if there is a good book that covers up that and also ...
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Conversion of mixed tensors into mixed tensors and into covariant (or contravariant) ones

I am an undergraduate student of Physics, currently taking a course on Special Relativity, but I am getting too confused with tensors and their indices. My question is: How to convert mixed tensors to ...
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Associativity of Relativistic Oblique Velocity Addition

I've encountered some information in the Wikipedia page on Lorentz transformation (https://en.m.wikipedia.org/wiki/Lorentz_transformation) that I am having difficulty reconciling with other ...
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1answer
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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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1answer
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Deriving a unique derivative operator with torsion such that $\triangledown g = 0$

From Wald's General Relativity, Chapter 3 problem 1c), in the solution from http://www.physics.drexel.edu/~dcross/papers/wald.pdf: $\triangledown g = 0$ becomes again $$\triangledown_a g_{bc} = \bar{...
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Relativistic Hamiltonian with Full Mass Matrix

In some application of Hamiltonian Monte Carlo one can provide a full mass matrix (metric tensor) for the kinetic energy in Hamiltonian equations: $$ K(p) = \frac{1}{2}p^TM^{-1}p$$ which reduces to ...
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Why are the fundamental and anti-fundamental representation in $SL(2,\mathbb{C})$ not equivalent? [duplicate]

I am currently learning symmetries/group theory and I learnt that the fundamental representation and the anti-fundamental representation of $SL(2,\mathbb{C})$ are not equivalent. This means that no ...
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Derivative of a scalar function with respect to a contravariant vector

So in the steps below, a scalar function, $f(x^2)$, is partial differentiated with respect to a contravarient vector: Where η is the metric tensor used for special relativity. I understand all the ...
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components of antisymmetric tensor unchanged under rotations proof

In "classical theory of fields" Landau states that the components of a completely antisymmetric tensor of a rank equal to the number of dimensions of the space remain unchanged after a rotation. I ...
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Explain these two non-relativistic (Newtonian) approximation

Here I get the generalization of Euler's equation for perfect fluids in special relativity: $$0=\mu(\frac{\partial u_i}{\partial t}+u^a\partial_au_i)+\frac{\partial p}{\partial x^i}+u_i\frac{\partial ...
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Sum of two velocities is smaller than the speed of light

Using the Lorentz transformation from special relativity, we get that the sum of two velocities can be expressed as $$u=\frac{u'+v}{1+\frac{u'v}{c^2}}.$$ Given that $|u'|,|v| \le c$, I want to prove ...
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1answer
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Car-garage paradox with just one door

Special relativity implies the possibility of some apparently paradoxical situations, which can ususally be made sense of if one applies the theory rigorously. One of these is the car-garage paradox: ...
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Origin of negative term $-ct^2$ in the Lorentz invariance of a 4-vector

So I am taking an introductory course in Special Relativity. In my book the spacetime 4-vector is defined as: $$X= \begin{bmatrix}ct \\ x \\ y \\ z \end{bmatrix}$$. Then the book proceeds to say that ...
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In Kronecker delta notation, what if any is the significance of using subscript or superscript indices?

Studying special relativity for an exam on an Italian book (Relatività - Barone for those interested), before introducing a geometrical analysis for Lorentz transformations and Minkowski spacetime, ...
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Power Series Expansion in Inverse Powers

In a book I am reading at the moment it says that the expression $$ H = \sqrt{P^2 c^2 + m^2 c^4} $$ can be "formally expanded in inverse powers of $c$" to obtain $$ H = mc^2 + \frac{1}{2m} P^2 + \...
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Confusion with notation of square bracket and round bracket of indices of a tensor

Refer to the following picture: I am confused with the last notation. Say if I got a tensor ${T^{abc}}_{de}$ and I would like to denote a new tensor which is defined by permuting the indices $a$ and $...
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1answer
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My problem with the velocity of the object with position $z=t^2-t^3$.

(a) (b)Using first derivative test, $$\frac{dz}{dt}=0.$$ We get $t=2/3$. When $t=2/3$ object move $z=4/27$ units far from $z=0$. (c) Velocity at which object departed from $z=0,$ $$\frac{dz}{dt}|_{...
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Rigorous formulation of special relativity.

I want to understand the theory of special relativity and I am reading from Resnick's Introduction to Special Relativity. But I want to prove everything rigorously. $\newcommand{\R}{\mathbf R}$ $\...
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28 views

Valid Interpretation of Special Relativity in terms of Length Contraction of Relative Distances

Suppose I have three points $O_1, O_2, O_3$ floating in a universe that obeys galilean relativity $O_2$ is located 10 metres east of $O_1$ (just pick a direction vector of your choice and call it ...
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101 views

reference for special relativity

I got a student finishing its first year of Math and she would like to study a bit of special relativity from a mathematics point of view. I know the subject quite well but I don't know any basic ...
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How to substitute an integral to proof covariance (special theory of relativity)

I encountered the following integral (with delta being the Dirac delta function): \begin{equation} \vec{J}^\prime(\vec{y})=\int{d\tau\delta(\vec{y}-\vec{x}^\prime(\tau))\frac{d\vec{x}^\prime}{d\tau}} \...
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Form of Lorentz Transformation in 3D space

My idea so far: I know that it must satisfy the relation: $$ L^T g L=g $$ where $g=diag(1,-1,-1,-1)$ hence I deduced from this it must preserve the quantity $$c^2t^2-x^2-y^2-z^2$$ I am unsure as to ...
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Special Relativity: A showing question $c^2-v^2$ using Velocity transformations

I have been trying to do this question for ages now: A particle has a velocity v ={$v_x,v_y,v_z$} in S and a velocity $\bf v'$={$ v'_x, v'_y,v'_z$} in S'. Prove from velocity transformations that $...
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Didn't understand a step in Einstein's paper on special relativity

I didn't understand a step in Einstein's paper(special relativity). Suppose we have a function $f(x', t)$ such that: $$\frac{1}{2}[f(0, t) + f(0, t + \frac{x'}{c - v} + \frac{x'}{c + v})] = f(x', t +...
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I am currently reading Special Relativity by Woodhouse, I need help with understanding divergence of magnetic fields

He states after recalling, $\textbf{B}=\frac{\mu_{0}e\textbf{v}\times\textbf{r}}{4\pi r^{3}}$, that $\frac{\textbf{r}}{r^{3}} = - grad(\frac{1}{r})$. First of all I am unsure how we comes to this ...
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Understanding the tensor notation in special/general theory of relativity

So I at the moment I am struggling with the tensor notation in special/general theory of relativity. So we had this in university: $$ \sum_{\nu = 0}^{3} \eta_{\mu \nu } \Lambda ^{\nu}_{\;\beta} = (...
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The reason of script expression of $\lambda^\mu_{\;\;\nu}$

I wonder why first making person who express $$\lambda^\mu_{\;\; \nu}$$ made like that? although $$\lambda^{\mu\nu},\lambda_{\mu\nu}, \lambda^{\;\;\mu}_\nu$$ are also possible!! please teach me ...
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Rocket to a ray of light

$A$ and $B$ are two stationary points on a line $30,000,000$ km apart. A light flashes at $B$, and at that precise moment a rocket takes off at $A$ at $180,000$ km/second. The rocket is considered ...
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What mathematical principal allows this rearrangement during simplifying

\begin{align} t^2 &= (t^I)^2-\frac{v^2}{c^2}(t^I)^2 \\ t^2 &= (t^I)^2\left(1-\frac{v^2}{c^2}\right) \end{align} I've been researching Einstein's Special Relativity Theory and would love to ...
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Series $\sum_{n=1}^{\infty} \left[K_0\left(\sqrt{[n\beta-it]^2+s^2 }\right)+K_0\left(\sqrt{[n\beta+it]^2+s^2}\right)\right]$

Let $\beta > 0$ and $t, s \in \mathbb{R}$. Furthermore, suppose that $-t^2 + s^2 > 0$. Define the following function: $$ F( \beta, t, s )\ : = \ \sum_{n=1}^{\infty} \left[K_0\left(\sqrt{[n\beta-...
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How to approximate this expression ?

Here 'n'is a constant number between 1 and 2 and we know that $V<<c$. Then how do you show that:$$\frac{1}{1+\frac{V}{nc}}$$ can be approximated by this expression:$$(1-\frac{V}{nc})$$
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Length Contraction [closed]

At what speed does a meter stick move if its length is observed to shrink to $0.5$ m? What would we think for this question? It doesn't seem too hard but couldn't work it out. That's why I want to ...
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Analysis of a function: Showing that Quantum Mechanics violates relativity

Consider the Hilbert space $L^2(\mathbb{R}^d)$ and a self-adjoint, positive operator $H$ (Hamiltonian). Let $\psi_t$ be a solution to the Schrödinger equation (with $\psi_0$ the initial condition), ...
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Lorentz-invariance Lagrangian for a free particle and $\frac{d}{d \tau} \left( m \eta_{\mu \nu} u^{\mu}\right)=0$

Considering a Lorentz-invariance Lagrangian for a free particle $$L=\frac{m}{2}\eta_{\nu \mu}u^{\mu}u^{\nu}$$ In the coordinates you use the Minkowski metric has constant components so the Euler-...
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Only at the vertex of the light-cone the vectors, that are tangent to the cone are all and only light-cone ones

Tangent vectors to a light-cone in the spacetime are not always vector timelike. The condition of tangency to a submanifolds is a linear equation in the tangent space, and this also applies to the ...
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$v^{i}= \frac{dx^{i}}{dt}=c\frac{dx^{i}}{d\tau}\left( \frac{dx^{0}}{d\tau} \right)^{-1}$ I do not understand where the last two equalities emerge

I'm considering a parametrization of the worldline $ \tau \longrightarrow x^{\mu}(\tau).$ Considering the projection of the universe line in three-dimensional space, how do speed components in three-...
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Deriving y'=y and z'=z from symmetry

Consider the following system It's clear that the Galilean transformation from the frame S to S' in this case is given by: $$\begin{align} x' &= x-vt \\ y' &= y \\ z' &= z \\ t' &= ...
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1answer
121 views

div(grad f ) with the Special Relativity Metric

I am looking at David Kay's "Tensor Calculus" 2011 Schaum's Outline; specifically equation 12.36 on page 172 and I am having difficulty making the leap from: $$ \text{div}(\text{grad}f) \equiv \Box f= ...
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135 views

Error range for the Taylor polynomial Lorentz factor $γ$

Consider the Lorentz factor (in special theory of relativity) as the function $$γ(x)=\frac{c}{\sqrt{c^2-x^2}},\;x\in[ 0 , c \rangle$$ Where $x=$ is the velocity of an object moving relative to ...
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108 views

Relativistic particle annihilation- getting wrong answer

While preparing for my exams, I found the following question on a past paper for which I am getting a different answer to what the question says I should be getting, I can't see where I am going wrong ...
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242 views

Show that $d^4k$ is Lorentz invariant

Show that $d^4k$ is Lorentz Invariant Relevant equations Under a lorentz transformation the vector $k^u$ transforms as $k'^u=\Lambda^u_v k^v$ where $\Lambda^u_v$ satisfies $\eta_{uv}\Lambda^{u}_{p}\...
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Relativistic sum with magnitude c

Pick any two vectors (in 3 dimensions) having magnitude equal to c and check whether the relativistic sum of them also has magnitude c. Is u v equal to v u?
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How to develop Einstein field equations for spherical symmetry space time?

Spherical Symmetry Space Time is given by: $ds^2$ = $e^\omega c^2dt^2$ - $e^\eta dt^2$ + $d\theta^2$+ $\sin^2 d\phi^2$ where $\eta$ and $\omega$ are functions of r. I have calculated all the Ricci ...
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216 views

Metric tensor vs metric space.

In special relativity the metric is expressed as $ds^2=-c^2dt^2+dx^2+dy^2+dz^2$ I have questions. 0- is this the metric tensor or the space metric? I am confused here, i.e. what is the ...