What is Lorentz transformation: Definition and 380 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 am on Chapter one and have not yet reached this formula yet but i wanted to know in advance because i was a bit curious
Also i noticed it written as (c²t²- x² ) in some places while (x² - c²t²) in some places ,is there a difference?
Hello everyone,
I recently have learned about space time intervals and how these intervals between two events are invariant across all inertial frames and this can be proven by using the full Lorentz transformation.
I wanted to learn more about the full Lorentz transformation and I read the...
Recollections of a late Spring semester's lesson describing the derivation of Lorentz's Transformation often solicit many unanswered questions. The textbook used has been secured; however, it is unknown. Whether, that secondary school instructor provided the premises used for the derivation from...
Are forces subject to the Lorentz transformation? Not force carriers; I already got that question answered, thanks to @PeterDonis. But forces. The different forms of them, such as the contact forces etc.: https://en.wikipedia.org/wiki/Contact_force
With time dilation, does the rate of force...
I made a tool for calculating and visualizing how the electric and magnetic fields transform under a Lorentz boost. Thought I'd share it here, in case anyone finds it interesting.
https://em-transforms.vercel.app/
The LT can be derived from the first postulate of SR, assuming linearity an that velocity composition is commutative, and that GT can be excluded: ##t' \neq t##.
Definition of the constant velocity ##v##:
##x' = 0 \Rightarrow x-vt=0\ \ \ \ \ \ ##(1)
With assumed linearity follows for the...
The Lorentz transform for velocities is as follows:
$$u=\frac{v+w}{1+\frac{vw}{c^{2}}}$$
But which assumption exactly underlies this so that you get exactly this formula and not any other formula with approximately the same properties?
While deriving Lorentz transformation equations, my professor assumes the following:
As ##\beta \rightarrow 1,##
$$-c^2t^2 + x^2 = k$$
approaches 0. That is, ##-c^2t^2 + x^2 = 0.## But the equation of the hyperbola is preserved in all inertial frames of reference. Why would ##-c^2t^2 + x^2##...
##\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}} =...
Two spaceships are heading towards each other on a collision course. The following facts are all as measured by an observer on Earth: spaceship 1 has speed 0.74c, spaceship 2 has speed 0.62c, spaceship 1 is 60 m in length. Event 1 is a measurement of the position of spaceship 1 and Event 2 is a...
Summary:: Special relativity and Lorentz Transformations - I got this problem from a first-semester course at university. I have been struggling for a few days and decided to get some help.
A rocket sets out from x = x' = 0 at t = t' = 0 and moves with speed u in the negative x'-direction, as...
Let there be a track 450,000 km long and a rocket 300,000 km long with a laser attached to the bottom of it's back end with a clock beside it, and a second synchronized clock attached to bottom of its front end. Both clocks were also synchronized with a track clock while the rocket was parked...
I believe this does not belong to the homework category. I hope I won't be mistaken.
I am reading a book to self-study special relativity, the following is an example mentioned in the book.
When clock C' and clock C1 meet at times t'=t1=0, both clocks read zero. The Observer in reference frame...
Is there the simplest, direct, and easy-to-understand method that only needs to apply the most basic algebra and logic to completely and strictly derive the Lorentz transformation?
Thanks for your help.
My textbook (from first year university physics) says that length contraction is actually real. But how can it be real when two different observers can measure two different lengths? For example, if I am in a spaceship going close to the speed of light relative to people on Earth, they will...
In the special theory of relativity, it seems impossible to derive the lorentz transformation without assuming that the lorentz factor is independent of the sign of the relative velocity. For some reason, I can't get my head around why this assumption is so easily made, as if it's trivial. Can...
Hi,
It's not homework but I still thought I better post it here.
Please have a look on the attachment. For hi-resolution copy, please use this link: https://imagizer.imageshack.com/img922/7840/CL6Ceq.jpg
I think in equations labelled "12", 'e' is electric charge and Ex is the amplitude of...
On the Yale University Prof Shankar Youtube vid 'Lorentz Transformation' Prof Shankar writes up on the board that x = ct and then x prime = c t prime.
It is the basis of all that follows. But i don't understand.
at x = 0, t = 0 and x prime = 0 and t prime = 0. He's got that written up...
Let me define ##L_{x;v}## as the operator that produce a Lorentz boost in the ##x##-direction with a speed of ##v##. This operator acts on the components of the 4-position as follows
$$L_{x;v}(x) =\gamma_{v}(x-vt),$$
$$L_{x;v}(y) =y,$$
$$L_{x;v}(z) =z,$$
$$L_{x;v}(t)...
In the Hamiltonian formalism, the space-time transformation are realized via canonical transformation, and the transformations are generated by Poisson brackets of certain functions of phase-space variables.
In Newtonian mechanics, Galilean boosts are generated by the sometimes called dynamic...
Hi,
I've read a number of posts here on PF about Einstein's clock synchronization convention.
In the context of SR we know the transformation law between inertial frame's coordinates is actually the Lorentz one. The invariant speed for Lorentz transformation is c (actually it coincides with...
The Lorentz transformations of electric and magnetic fields (as given, for example in Wikipedia) are
$$
\begin{align*}
\bar{\boldsymbol{E}}_{\parallel} & =\boldsymbol{E}_{\parallel}\\
\bar{\boldsymbol{E}}_{\perp} &...
Hello again. I am sorry I got another problem when learning QFT regarding the Lorentz transformation of derivatives.
In David Tong's notes, he says
I am not sure how to transform the partial derivatives. Explicitly, should ##\frac {\partial} {\partial x ^{\mu}}## transform to ##\frac...
This approach is seeming intuitive to me as I can visualize what's going on at each step and there's not much complex math. But I'm not sure if I'm on the right track or if I'm making some mistakes. Here it is:
##A## has set up a space-time co-ordinate system with some arbitrary event along his...
a) I think I got this one (I have to thank samalkhaiat and PeroK for helping me with the training in LTs :) )
$$\eta_{\mu\nu}\Big(\delta^{\mu}_{\rho} + \epsilon^{\mu}_{ \ \ \rho} +\frac{1}{2!} \epsilon^{\mu}_{ \ \ \lambda}\epsilon^{\lambda}_{ \ \ \rho}+ \ ...\Big)\Big(\delta^{\nu}_{\sigma} +...
This exercise was proposed by samalkhaiat here
Given the defining property of Lorentz transformation \eta_{\mu\nu}\Lambda^{\mu}{}_{\rho}\Lambda^{\nu}{}_{\sigma} = \eta_{\rho \sigma}, prove the following identities
(i) \ (\Lambda k) \cdot (\Lambda x) = k \cdot x
(ii) \ p \cdot...
Dear reader,
there is a physics problem where I couldn't understand what the solutions.
It is about the lorentz transformation of a bilinear spinor matrix element thing.
So the blue colored equation signs are the parts which I couldn't figure out how.
There must be some steps in between which...
As we all know, for the reference frame S' and S of relative motion, according to Lorentz transformation, we can get
As we all know, for the reference frame S' and S of relative motion, according to Lorentz transformation, we can get
As we all know, for the reference frame S' and S of relative...
I wanted to make a derivation of the Lorentz transformation :
$$x'=Ax+Bt\\t'=Dx+Et$$
The conservation of the quadratic form ##c^2t'^2-x'^2## yields the equations:
$$A^2-B^2/c^2=1\\D^2-E^2/c^2=-1/c^2\\AD=BE/c^2$$
Hence ##B=c\sqrt{E^2-1}##,##D=\sqrt{E^2-1}/c##,##A=\pm E##.
The speed of the...
What is the difference between special relativity and the Lorentz transformation? Aren't they basically the same thing?
Also, I was wondering what about matter makes spacetime curve?
If we have motion of system ##S'## relative to system ##S## in direction of ##x,x'## axes, Lorentz transformation suppose that observers in the two system measure different times ##t## and ##t'##.
x'=\gamma(x-ut)
x=\gamma(x'+ut')
Why we need to use the same ##\gamma## in both relations? Why not...
Homework Statement: This seemed at first glance very easy. But there appeared some confusion.
A is moving to the right with velocity v with respect to B. The proper time for A is ##t_a=t_b\sqrt{1-v^2/c^2}##. And B is moving to the right with velocity u with respect to C. Proper time for B...
[BEGINNGING NOTICE]
Before I begin showing my attempted solution, I would just like to quickly mention that this is a "repost" of the same question I had around a week ago. While I would usually use the "reply" function on the same thread, I believe that thread is getting pretty messy (sometimes...
"My" Attempted Solution
To begin, please note that a lot - if not all - of the "solution" is largely based off of @eranreches's solution from the following website: https://physics.stackexchange.com/questions/369352/scalar-invariance-under-lorentz-transformation.
With that said, below is my...
Summary: The problem is to generalize the Lorentz transformation to two dimensions.
Relevant Equations
Lorentz Transformation along the positive x-axis:
$$ \begin{pmatrix}
\bar{x^0} \\
\bar{x^1} \\
\bar{x^2} \\
\bar{x^3} \\
\end{pmatrix} =
\begin{pmatrix}
\gamma & -\gamma \beta & 0 & 0 \\...
with distance between planets as 4x10^8m measured by you on the ship
My attempt:
t' = γ(t - ux/c^2)
γ = 5/3
u = 0.8c
t = 0.9s
x = 4x10^8m
answer is: -0.278
Therefore not possible
My question is what if we traveled rightwards, from p2 to p1, would the answer change?
Should my above information...
Unfortunately, I am not entirely confident of the above equations being able to do the trick and ultimately solve for the question. However, my guess is that using the equation written above for "boost", I could perhaps use ##v## and insert it into the ##x##-direction part of the matrix...
I have been getting back to studying physics after a long break and decided to go through the problems in Rindler. But there is something I don't quite understand in this problem.
To first answer the second part, Exercise II(12), I wrote $$\frac{du_2}{dt} = \frac{du_2}{du_2^\prime}...
I am trying to understand the last block of equations in the picture (after 3.31). In the first line of that block, he transforms the spinor ##\psi## which I have no problem with. What I have a problem with is the ##\gamma ^{\mu} \partial _{\mu}##. They form a Lorentz scalar, so they should not...
I want to know why an else solution can not get the right answer. And want to know the way to correct this solution.Supposed that a frame S'' is moving in the lab frame at ##\beta_x## in the x-direction, ##\beta_y## in the y-direction, now I want to find out the Lorentz transformation between...
A particle is moving in the lab frame ##S'## at ##\beta'_z##. I want to transform coordinates and momenta of the particle to a frame ##S## moving at ##\beta_0##.
At time ##t = t' = 0##:
$$z = \frac{z'} { \gamma_0 (1 - \beta'_z \beta_0) },\,
\gamma\beta_z = \gamma_0 ( \gamma'\beta'_z -...
I'm currently watching lecture videos on QFT by David Tong. He is going over lorentz invariance and classical field theory. In his lecture notes he has,
$$(\partial_\mu\phi)(x) \rightarrow (\Lambda^{-1})^\nu_\mu(\partial_\nu \phi)(y)$$, where ##y = \Lambda^{-1}x##.
He mentions he uses active...
I would like to apply a General Lorentz Boost to some Multi-partite Quantum State.
I have read several papers (like this) on the theory of boosting quantum states, but I have a hard time applying this theory to concrete examples.
Let us take a ##|\Phi^+\rangle## Bell State as an example, and...
I read the Lorentz transformation can be obtained by solving the requirement of invariance of the wave equation. If one considers linear transformations this the same as the spacetime interval squared to be invariant.
What are the other nonlinear transformations keeping the wave equation...
Problem Statement: Consider three frames Σ (x, y, z, t), Σ' (x', y', z', t'), and Σ'' (x'', y'', z'', t'') whose x, y, and z axes are parallel at each point in time stay. Σ' moves relative to Σ with velocity v1 along the x-axis. The system Σ'' moves relative to Σ' with the velocity v2 along the...
Hi guys, I'm reading a book 'the theoretical minimum: special relativity and classical field theory'. In chapter 1.3, author explains the general Lorentz transformation.
He said "Suppose you have two frames in relative motion along some oblique direction, not along any of the coordinate axes...