Lorentz transformations Definition and 175 Threads

Hi, reading again this old thread about Index notation for inverse Lorentz transform, I believe there is a missing ##\hat{L}## in the following, namely $$(\hat{\eta} \hat{L} \hat{\eta})^{\text{T}} \hat{L}=\hat{\eta} \hat{L}^{\text{T}} \hat{\eta} = \mathbb{1} \; \Rightarrow \; \hat{L}^{-1} =...

Hi, starting from a recent thread, I'd like to discuss a point related to the Lorentz transformation of EM Faraday tensor field between inertial reference frame. As explained in this video at minute 11:20, in the Lab inertial frame there is only a magnetic field B in the region surrounding the...

I've arrived to an expected answer, but I am not sure at all that the process was what the problem statement wants. First, I considered ##0=(t+\delta t)^2-(x+vt)^2-(t^2-x^2) \approx 2t \delta t - 2xvt - v^2t^2##. Ignoring ##O(v^2)## gives ##\delta t=vx##, i.e., ##t \rightarrow t+vx##. Keeping...

Consider the Lorentz transformations with c=1, and consider any point in space whose x coordinate isn't zero, starting from ##t_{inital }= t'_{inital }=0## ##t' =\gamma (t-xv)## ##t= \frac { t'}{\gamma} + xv## ##\Delta t' = t'-0## ##\Delta t = t-0## Time dilation provides ##\Delta t' =\gamma...

I was reading Einstein's Simple Derivation of the Lorentz Transformation which is Appendix I in his book Relativity: the Special & the General Theory. (Online copies can be found at Bartleby's and the Gutenberg Project websites.) I came across an interesting but confusing result by using the...

Could one derive a set of coordinate transformations that transforms events between different reference frames in the de Sitter metric using the invariant line element, similar to how the Lorentz Transformations leave the line element of the Minkowski metric invariant? Would these coordinate...

In the frame of Observer C standing by the side of the road, the speed of Car A with respect to Car B = v1 + v2. (Galilean Transformation). In the frame of Car A, the speed of Car B < v1 + v2 (Lorentz Transformation). Please tell me if this understanding is correct.

##\bar{\mathcal{O}}## is moving with a velocity ##v## relative to ##\mathcal{O}## along ##x^{1}## The Lorentz transformations between a Frame ##\mathcal{O}## and ##\bar{\mathcal{O}}## is given by: $$\Delta x^{\bar{0}} = \gamma\left(\Delta x^0 - v\Delta x^1\right)$$ $$\Delta x^{\bar{1}} =...

Hi, I´m trying to solve a special relativity problem, and I think I need some help. There are two inertial frames of reference, ##O## and ##O'##, the last one moving with relative velocity ##v## in the ##x## direction. There's a rod with length ##L'## fixed to frame ##O'##, such that front end...

A π+ meson is an elementary particle with a mean lifetime, defined in its rest frame, of τ = 2.60×10−8 s. The meson decays to a muon (µ+) and a neutrino (νµ) via the reaction π+ → µ+ + νµ. A π+ traveling in the laboratory decays so that the µ+ travels in the same direction as the original π+ and...

For a complex null tetrad ##(\boldsymbol{m}, \overline{\boldsymbol{m}}, \boldsymbol{l}, \boldsymbol{k})##, how to arrive at formulae (3.14), (3.15) and (3.17)? The equation (3.16) is clear as is. (I checked already that they work i.e. that ##\boldsymbol{e}_a' \cdot \boldsymbol{e}_b' = 2m'_{(a}...

I consider three material points O, O', M, in uniform rectilinear motion in a common direction, so that in relation to the point O, the points O' and M move in the same direction with the constant velocities v and u (u>v>0). Assuming that at the initial moment (t0=0), the points O, O', M were in...

Hi, I was looking at this derivation https://en.wikipedia.org/wiki/Derivations_of_the_Lorentz_transformations#From_group_postulates and I was wondering 1- where does the group structure come from? The principle of relativity? or viceversa? or what? 2- why only linear transformations? I remember...

Hello, why time is the fourth dimention and not another quantity or variable? General relativity has as a special case the special relativity, so Lorentz transformations are contained in general relativity but are they in a more general form than that of special relativity generally? If they...

[Mentors' note: This thead was forked from another thread - hence the reference to "these replies" in the first post] I am wondering why all these replies only discuss Lorentz transformations in 1+1 spacetime dimensions. That is the easy bit. The problems in understanding arise in 2+1...

The Lorentz tranformations are: ##x' = \gamma (x-vt) ## ##t' = \gamma(t - \frac{vx}{c^2})## Consider an event (x,t) happening in S frame. Let S' frame be moving w.r.t. S frame along x direction with speed v whose origins coincide at t=0. We find that the new coordinates of this event are...

Good evening, I'm trying to solve this exercise that apparently seems trivial, but I wouldn't want to make mistakes, just trivial. Proceeding with the first point I wonder if my approach can be correct in describing this situation. From the assumptions, the two fields are in this...

I started by finding the main events: Sending the first message Receipt the first message Sending the second message Receipt the second message Now, what we know is the time by ##S'## (comoving frame with the spaceship) ##T_1'## and ##T_2'## remaining to arrive to the Earth measured at...

Before to open this topic, I found this there. It's quite similar, if not the same, but I'm a little confused, so I'm here. The situation is represented in this image. From optical geometry, ##\theta_{incident} = \theta_{reflected}## The four-momentum in ##S'## is the following one...

We take an arbitrary spacetime point ##(x,t)## in any observer's reference frame ##A##. Let ##(x(v),t(v))## be the co-ordinates of this same event as seen from a frame ##B## moving at a velocity ##v## wrt ##A##. As ##v## varies, the set of points ##(x(v),t(v))## constitute some curve ##C##. So...

I am totally new to the theory of Special Relativity, but find it very facinating. As a young man I saw a few documentaries on how Einstein saw a clock's movement reaching noon, and how he, traveling in a tram heard the gong only later. He then thought about what if he traveled at the speed of...

I'm trying my hand at deriving Lorentz transformations using 3 postulates - it's a linear transformation, the frames are equivalent, so they see the same speed of each other's origins and that the speed of light is the same. Let's say frame ##S## is moving at velocity ##v## in the...

I am reading Tong's lecture notes and I found an example in which there are several aspects I do not understand. This example is aimed at: - Understanding what is the analogy in field theory to the fact that, in classical mechanics, rotational invariance gives rise to conservation of angular...

Could it be that the transformations keeping the wave equation invariant have other solutions than the usual Lorentz ones ?

Summary: The problem is to generalize the Lorentz transformation to two dimensions. Relevant Equations Lorentz Transformation along the positive x-axis: $$ \begin{pmatrix} \bar{x^0} \\ \bar{x^1} \\ \bar{x^2} \\ \bar{x^3} \\ \end{pmatrix} = \begin{pmatrix} \gamma & -\gamma \beta & 0 & 0 \\...

In his book, Landau derives the Lorentz transformations using the invariance of the interval, and I have some questions about it that I would like to clarify 1. What is a parallel displacement of a coordinate system? Does it refer to moving along any axis? I don't see how any arbitrary...

Two frames measure the position of a particle as a function of time: S in terms of x and t and S', moving at constant speed v, in terms of x' and t'. The acceleration as measured in frame S is $$ \frac{d^{2}x}{dt^{2}} $$ and that measured in frame S' is $$ \frac{d^{2}x'}{dt'^{2}} $$My question...

There are several ways to show that the Lorentz transformations must be linear. What's the best/more intuitive argument in your opinion?

It seems that there is a considerable number of ways of deriving the Lorentz transformations. Does anyone know how many ways are there?

In the Earth’s reference frame, a tree is at x=0km and a pole is at x=20km. A person stands at x=0 (stationary relative to the Earth), and at t=10 microseconds, this person witnesses two simultaneous lightning strikes. One of these strikes hits the tree he is standing under, and the other hits...

I understand x' = λ(x - vt) but why does t' = λ(t - vx/c^2)? where does the vx/c^2 come from? and honestly I don't understand what t' is. because from what I understand is that t' is the length of time t as observed from the reference frame S'. which means t' = t*λ?

I have tried to derive the lorentz transformations but there is a part of it that requires substitution into two equations when t=0. How do I do that

In my last post I asked about the general form of the Lorentz Transformation for time. Now I am trying to understand the final form of it, and how it makes sense based on what's happening physically. The final form for t is: t = γt1 + (γv/c2/)x1 It's the second part of this equation, the...

I am trying to understand the general form of the Lorentz Transformations before I even get into the long process of deriving that into the specific equations. In Taylor and Wheeler's, Spacetime Physics book they give this as the general form: t= Bx1 + Dt1 x= Gx1 + Ht1 In the equation for t...

Hi everyone! I have a problem with one thing. Let's consider the Lorentz group and the vicinity of the unit matrix. For each ##\hat{L}## from such vicinity one can prove that there exists only one matrix ##\hat{\epsilon}## such that ##\hat{L}=exp[\hat{\epsilon}]##. If we take ##\epsilon^{μν}##...

I have just started learning the Special Theory of Relativity. While deriving, I am facing some problems. I obviously have made some kind of mistake while using the equations... What is wrong if I don't use the time transformation equation in Event #2?

Homework Statement I have a mother particle at rest, which decays to a daughter particle. The daughter has mass m, momentum p and energy E and is at an angle θ1. Now I have to assume that the daughter is emitted at an angle θ2, and the mother is moving along the x-axis with velocity βc. I need...

Homework Statement Special Relativity Question. Consider objects 1 and 2 moving in the lab frame; they both start at the origin, and #1 moves with a speed u and #2 moves with a speed v. They both move in straight lines, with an angle θ between their trajectories (again in the lab frame). What...

Are these two subjects closely related? It seems a tensor can be invariant when viewed from any **co ordinate system and The Lorentz Transformation seems to allow 2 moving co ordinate frames to agree on a space time intervals. Is there some deep connection going on? **=moving frames of...

I've been working my way through Peskin and Schroeder and am currently on the sub-section about how spinors transform under Lorentz transformation. As I understand it, under a Lorentz transformation, a spinor ##\psi## transforms as $$\psi\rightarrow S(\Lambda)\psi$$ where...

Let ##\Lambda## be a Lorentz transformation. The matrix representing the Lorentz transformation is written as ##\Lambda^\mu{}_\nu##, the first index referring to the rows and the second index referring to columns. The defining relation (necessary and sufficient) for Lorentz transforms is...

Homework Statement 2. The attempt at a solution 3. Relevant equations In the first problems of that book i was using the Galilean transformations where V1 = V2 + V But if i use that then V1 = 0.945 - 0.6 V1 = 0.345 Is not the same result, so I am confused. In this new problems we are...

Homework Statement Show how one can obtain the Doppler transformation for the frequency of a receding source just using the Lorentz transformations for the energy (where E=h). Homework Equations Relativistic transformations for momentum and energy: E = γ(E' + vp'x) pc/E = v/c = β The Attempt...

I was reading my textbook for my elementary modern class and the author said that a pulse of light from a light bulb would be spherical and could be expressed as x2 + y2 + z2 = c2t2 and x'2 + y'2 + z'2 = c2t'2. Then the author goes on to say that this cannot happen for both reference frames in a...

In my simulation of the twin paradox, i used the Lorentz transformation formulas to map events from one inertial reference frame into another IRF. Reading through various threads here, i read that spacetime is curved and that space can be considered flat only for small distances. So my...

Hello everyone, There is something that has been bugging me for a long time about the meaning of Lorentz Transformations when looked at in the context of tensor analysis. I will try to be as clear as possible while at the same time remaining faithful to the train of thought that brought me...

Ok so... It's been a while since I first saw this problematic scenario and I want to know how to deal with it. The question arises in the context of special relativity. Suppose 2 objects moving at the same speed. The floor is the rest frame 'A' and the front object is the moving frame 'B'. The...

Homework Statement Show that an infinitesimal boost by v^j along the x^j-axis is given by the Lorentz transformation \Lambda^{\mu}_{\nu} = \begin{pmatrix} 1 & v^1 & v^2 & v^3\\ v^1 & 1 & 0 & 0\\ v^2 & 0 & 1 & 0\\ v^3 & 0 & 0 & 1 \end{pmatrix} Show that an infinitesimal rotation by theta^j...

In special relativity, the electromagnetic field is represented by the tensor $$F^{\mu\nu} = \begin{pmatrix}0 & -E_{x} & -E_{y} & -E_{z}\\ E_{x} & 0 & -B_{z} & B_{y}\\ E_{y} & B_{z} & 0 & -B_{x}\\ E_{z} & -B_{y} & B_{x} & 0 \end{pmatrix}$$ which is an anti-symmetric matrix. Recalling the...

Homework Statement I am meant to show that the following equation is manifestly Lorentz invariant: $$\frac{dp^{\mu}}{d\tau}=\frac{q}{mc}F^{\mu\nu}p_{\nu}$$ Homework Equations I am given that ##F^{\mu\nu}## is a tensor of rank two. The Attempt at a Solution I was thinking about doing a Lorents...