What is Lorentz transformations: Definition and 173 Discussions
In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation is then parameterized by the negative of this velocity. The transformations are named after the Dutch physicist Hendrik Lorentz.
The most common form of the transformation, parametrized by the real constant
v
,
{\displaystyle v,}
representing a velocity confined to the x-direction, is expressed as
t
′
=
γ
(
t
−
v
x
c
2
)
x
′
=
γ
(
x
−
v
t
)
y
′
=
y
z
′
=
z
{\displaystyle {\begin{aligned}t'&=\gamma \left(t-{\frac {vx}{c^{2}}}\right)\\x'&=\gamma \left(x-vt\right)\\y'&=y\\z'&=z\end{aligned}}}
where (t, x, y, z) and (t′, x′, y′, z′) are the coordinates of an event in two frames, where the primed frame is seen from the unprimed frame as moving with speed v along the x-axis, c is the speed of light, and
γ
=
(
1
−
v
2
c
2
)
−
1
{\displaystyle \gamma =\textstyle \left({\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\right)^{-1}}
is the Lorentz factor. When speed v is much smaller than c, the Lorentz factor is negligibly different from 1, but as v approaches c,
γ
{\displaystyle \gamma }
grows without bound. The value of v must be smaller than c for the transformation to make sense.
Expressing the speed as
β
=
v
c
,
{\displaystyle \beta ={\frac {v}{c}},}
an equivalent form of the transformation is
c
t
′
=
γ
(
c
t
−
β
x
)
x
′
=
γ
(
x
−
β
c
t
)
y
′
=
y
z
′
=
z
.
{\displaystyle {\begin{aligned}ct'&=\gamma \left(ct-\beta x\right)\\x'&=\gamma \left(x-\beta ct\right)\\y'&=y\\z'&=z.\end{aligned}}}
Frames of reference can be divided into two groups: inertial (relative motion with constant velocity) and non-inertial (accelerating, moving in curved paths, rotational motion with constant angular velocity, etc.). The term "Lorentz transformations" only refers to transformations between inertial frames, usually in the context of special relativity.
In each reference frame, an observer can use a local coordinate system (usually Cartesian coordinates in this context) to measure lengths, and a clock to measure time intervals. An event is something that happens at a point in space at an instant of time, or more formally a point in spacetime. The transformations connect the space and time coordinates of an event as measured by an observer in each frame.They supersede the Galilean transformation of Newtonian physics, which assumes an absolute space and time (see Galilean relativity). The Galilean transformation is a good approximation only at relative speeds much less than the speed of light. Lorentz transformations have a number of unintuitive features that do not appear in Galilean transformations. For example, they reflect the fact that observers moving at different velocities may measure different distances, elapsed times, and even different orderings of events, but always such that the speed of light is the same in all inertial reference frames. The invariance of light speed is one of the postulates of special relativity.
Historically, the transformations were the result of attempts by Lorentz and others to explain how the speed of light was observed to be independent of the reference frame, and to understand the symmetries of the laws of electromagnetism. The Lorentz transformation is in accordance with Albert Einstein's special relativity, but was derived first.
The Lorentz transformation is a linear transformation. It may include a rotation of space; a rotation-free Lorentz transformation is called a Lorentz boost. In Minkowski space—the mathematical model of spacetime in special relativity—the Lorentz transformations preserve the spacetime interval between any two events. This property is the defining property of a Lorentz transformation. They describe only the transformations in which the spacetime event at the origin is left fixed. They can be considered as a hyperbolic rotation of Minkowski space. The more general set of transformations that also includes translations is known as the Poincaré group.
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...
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...
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?
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...
Homework Statement
Spaceship A of length 30m travels at 0.6c past spaceship B. Clocks in frame S' of spaceship A and S of spaceship B are synchronised within their respective frames of reference and are set to zero, so that t' = t = 0 at the instant the front of spaceship A passes the rear of...
Hey guys,
In what circumstance or scenario would you use Lorentz transformations as a opposed to time dilation or length contraction? The reason that I ask this is because in all of the problems that I have worked with, the observer is always stationary relative to the event. For example, if...