What is Lorentz transformation: Definition and 379 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.
Hi,
I have trouble understanding why the following relations hold true. Given the Minkowski metric \eta_{\alpha\beta}=diag(1,-1,-1,-1) and the line segment ds^2 = dx^2+dy^2+dz^2, then how can i see that this line segment is equal to ds^2 = \eta_{\alpha\beta}dx^\alpha dx^\beta . Further, we...
Homework Statement
For an event occurring at (x,t),
consider the quantity I = x^2 - (ct)^2
Find a simple expression for this in the S' frame: I' = x'^2 - (ct')^2
How are I and I' related, and why is this noteworthy?
The Attempt at a Solution
So the question is under "Lorentz Transformation"...
Homework Statement
[/B]
A spaceship is approaching Earth from the far side of the sun. The Earth and sun are 8 light minutes apart and the ship is traveling at .8c. Two events are indisputable. 1) the ship is at the sun 2) the ship is at the earth. Assume that the Earth and sun are at rest...
Homework Statement
A rocket is traveling toward a galaxy with speed v.
a) If NASA says that distance from Earth to the galaxy is d, what is the distance d' from Earth to the galaxy according to the astronauts?
b) The astronauts experience a travel time to the galaxy t' and NASA records the...
For quite a long time now I'm having some trouble to bridge the gap between two different approaches to Special Relativity.
The first approach is the traditional one. It is the approach that Einstein presented in his paper and that is taught in most of the basic textbooks. In this approach...
Hi. First, excuse my English.
In my lecture notes on classical electrodynamics, we are introduced to the Lorentz transformations: a system S' moves relative to a system S with positive veloticy v in the x-axis (meassured in S), spatial axis are parallel, origin of times t and t' coincide...
In a standard problem of an electron released from the negative plate in an E field between 2 parallel plates in which the velocity must be determined why can the Lorentz transformation be used (involving v^2/c^2) when the electron is undergoing acceleration and there is nothing in the...
In Einstein’s book Relativity – the special and the general Theory (authorized translation by Robert W. Lawson, University of Sheffield) in chapter XI (the Lorentz Transformation), he gives us these formulas as the transforms:
x’ = (x-vt)/sqr(1-(v^2/c^2))
y’ = y
z’ = z
t’ = t-(v/c^2) ∙x /...
They seem to defy the most fundamental principle of SR. The first postulate/equivalence principle.
According to wikipedia, we get
Lorentz boost (x direction)
and slightly different formulas for the inverse Lorentz boost
"This "trick" of simply reversing the direction of relative velocity...
Let's say you have a rod that is 10 meters long. Observer O sees the ends of the rod at (t=0, x=0), and (t=0, x=10). Observer O' moves at speed v = 0.8c relative to O. What is the length of the rod in O's perspective?
Using the length contraction formula L' = γL, we find that O' sees the rod as...
Homework Statement
Using the tensor transformation law applied to ##F_{\mu\nu}##, show how the electric and magnetic field ##3##-vectors ##\textbf{E}## and ##\textbf{B}## transform under
(a) a rotation about the ##y##-axis,
(b) a boost along the ##z##-axis.
Homework Equations
The Attempt at...
Homework Statement
[/B]
A Lorentz transformation ##x^{\mu} \rightarrow x'^{\mu} = {\Lambda^{\mu}}_{\nu}x^{\nu}## is such that it preserves the Minkowski metric ##\eta_{\mu\nu}##, meaning that ##\eta_{\mu\nu}x^{\mu}x^{\nu}=\eta_{\mu\nu}x'^{\mu}x'^{\nu}## for all ##x##. Show that this implies...
Homework Statement
Show that the isotropy and homogeneity of space-time and equivalence of different inertial frames (first postulate of relativity) require that the most general transformation between the space-time coordinates (x, y, z, t) and (x', y', z', t') is the linear transformation...
I recently started studying Special Relativity an my book discusses the following:
Say I have synchronized two separated clocks in a reference frame S, if then an observer in another reference frame S' for whom the clocks are moving sees the clocks he would say those clocks are out of...
I don't understand why we can write the elements of the lorentransformation in the form
## {\Lambda}^{\mu}\:_{\nu} = [exp(-\frac{i}{2}{\omega}^{\rho\sigma}M_{\rho\sigma})]^{\mu}\:_{\nu} ##
I know that we can write it in the form
## {\Lambda} = exp(t\Theta) ##
where
## \Theta ##
are elements...
Are you good with Lorentz' tranformations ?
I tought I was, until I tried to do this exercice (it is really classical):
Two planets, A and B, are at rest with respect to each other, a distance L apart, with synchronized clocks. A spaceship flies at speed v past planet A toward planet B and...
Homework Statement
Derive the Lorentz Transformation using light cone coordinates defined by
##x^±=t±x##
##x^+ x^-~## is left invariant if we multiply ##~e^φ~## to ##~x^+~## and ##~e^{-φ}~## to ##~x^-~##, that is ##~x'^+ x'^-=x^+ x^-##
Homework Equations
##t'^2 - x'^2 = t^2 - x^2...
Hi,
I've been studying Dirac's theory of fermions. A classic topic therein is the proof that the equation is covariant. Invariably authors state that the gamma-matrices have to be considered constants: they do not change under a Lorentz-transformation. I am looking for the reason behind this. It...
Hi. In the attached proof for Lorentz transformation for momentum http://www.colorado.edu/physics/phys2170/phys2170_sp07/downloads/lorentz_transformation_E_p.pdf, there is this step that I don't understand:
1/√1-u'2/c2 = γ(1-vux/c2)/√1-u2/c2
Can someone explain how they derived this? Thanks! :)
Taking a look at "http://www.space.com/30026-earth-twin-kepler-452b-exoplanet-discovery.html" I observe that planet Kepler-452b (judged to be somewhat Earth-like) is 1400 light years from Earth. If a spaceship leaves Earth at a fifth of the speed of light, traveling toward Kepler-452b, from...
I'm going through Ray D'Iverno's "Introducing Einstein's Relativity", and there is a step he makes in deriving the Lorentz transformations that doesn't seem necessary to me. So I'm not sure what I'm missing. He derives them from Einsteins postulates of relativity. From the postulate that the...
Homework Statement
We now specify the velocity v to be along the positive x1-direction in S and of magnitude v. We also consider a frame \overline{S} which moves at speed u with respect to S in the positive x1-direction.
question 1 : Write down the transformation law for p^\mu .
question 2...
The thought experiment used to prove Lorentz transform uses a light signal as an assumption. What if there was something other than the light signal then Lorentz transformation would have totally different term in place of 'c'(speed of light).
Lorentz contraction problem:
By Bertrand Boucquillon
Components of the problem:
- Bob (observer)
- 2 identical rods that both measure 1 meter. Let's call them rod X and rod Y
- Point A
- Point B
Scenario (step by step):
1) Bob is at point A, and is at rest with both rods in his hands
2) Bob...
I understand that in order to preserve the inner product of two four vectors under a change of coordinates x^{\mu}\rightarrow x^{\mu^{'}}=\Lambda^{\mu^{'}}_{\,\, \nu}x^{\nu} the Minkowski metric must transform as \eta_{\mu^{'}\nu^{'}}=\Lambda^{\alpha}_{\,\...
Homework Statement
Is the transformation ##Y:(t,x,y,z)\rightarrow (t,x,-y,z)## a Lorentz transformation? If so, why is it not considered with P and T as a discrete Lorentz transformation? If not, why not?
Homework Equations
The Attempt at a Solution
A Lorentz transformation ##\Lambda##...
Hi All;
I was trying to understand Lorentz Transformation equation and special theory of relativity, but as I compared the derivation with a thought experiment which I imagined I found the whole Lorentz Transformation Equation fails. The details of the problem is given below. I know I m wrong...
Hi All;
I was trying to understand Lorentz Transformation equation and special theory of relativity, but as I compared the derivation with a thought experiment I created I found the whole Lorentz Transformation Equation fails. The details of the problem is given in the pdf file attached. I know...
Homework Statement
There are three observers, all non accelerating. Observer B is moving at velocity vBA with respect to observer A. Observer C is moving at velocity vC B with respect to observer B. All three observers and all their relative velocities are directed along the same straight line...
Hi, the following is taken from Peskin and Schroeder page 36:
##\partial_{\mu}\phi(x) \rightarrow \partial_{\mu}(\phi(\Lambda^{-1}x)) = (\Lambda^{-1})^{\nu}_{\mu}(\partial_{\nu}\phi)(\Lambda^{-1}x)##
It describes the transformation law for a scalar field ##\phi(x)## for an active...
Homework Statement
I have a particle moving with uniform velocity in a frame ##S##, with coordinates $$ x^\mu , \mu=0,1,2,3. $$
I need to show that the particle also has uniform velocity in a frame ## S' ##, given by
$$x'^\mu=\dfrac{A_\nu^\mu x^\nu + b^\mu}{c_\nu x^\nu + d}, $$
with ##...
I will start with a summary of my confusion: I came across seemingly contradictory transformation rules for left and right chiral spinor in 2 books, and am unable to understand what part is Physics and what part is convention. Or is it that one of the two books incorrectly writes the...
Hi there, kinda new here so please let me know if this question has been answered. I am hoping to get a link or two to some good sources of information on Lorentz transforms and distribution functions (as used in physics). I understand DF's in class and I understand the math behind them I just...
1,2,3. Homework Statement
I tried to derive the length contraction using the Lorentz transformation matrix and considering 2 events. I reached the correct result but there's a step that I had to assume that I don't understand.
Consider a ruler of length L along the x-axis for an observer at...
Given the Lorentz matrix Λuv its transpose is Λvu but what is its transpose ? I have seen ΛuaΛub = δb a which implies an inverse. This seems to be some sort of swapping rows and columns but to get the inverse you also need to replace v with -v ? Also In the LT matrix is it the 1st slot...
Hello,
I'm reading Tong's lecture notes on QFT and I'm stuck on the following problem, found on p.11-12.
A scalar field \phi , under a Lorentz transformation, x \to
\Lambda x , transforms as
\phi(x) \to \phi'(x) = \phi(\Lambda^{-1} x)
and the derivative of the scalar field transforms...
Hello,
a derivation of the lorentz transformation for an arbitrary direction of the relative velocity often makes use of writing the spatial position vector of an event as the sum of its component parallel and perpendicular to the velocity vector in one inertial frame and then transforming both...
I want to learn how to write down a particle state in some inertial coordinate frame starting from the state ##| j m \rangle ##, in which the particle is in a rest frame.
I know how to rotate this state in the rest frame, but how does one write down a Lorentz boost for it? Note that I am not...
Dear PF Forum,
First, I'd like to thanks this forum for helping this much and so far.
I have a question about Lorentz Transformation. Lots of questions actually :smile:
http://en.wikipedia.org/wiki/Lorentz_transformation#Boost_in_the_x-direction
Instead of using t and x, I'd like to use ta and...
Homework Statement
Light (plane wave) reflects from the mirror moving along X-axis with speed V. The wave is orthogonal to the mirror (φ=0°).
Write the law for frequency change.
Homework Equations
I know Lorenz transformation for frequency.
The Attempt at a Solution
All I do not know is how to...
Hello,
I have some mathematics background but little to no physics background. I am very interested in physics and am beginning to learn about relativity. Upon exploring the derivation of the Lorentz Transformation equation I noticed something that confused me a little. Again, I don't have much...
t' = γ(t - vx/c2)
where t is the time in the stationary frame
t' is the time in the moving frame
v is the relative velocity between the two frames
and x is the distance traveled in the time t in the stationary frame.
∴ x = vt
substituting this into the Lorentz time equation gives us:
t' = γt(1 -...
Hi All,
I am trying to understand the directional independence of LT. In contrast, we all know that Doppler Effect is dependent on the direction of motion. I have tried to find any reasoning or explanation and could not find one so far. May be I did not use correct terms in my searches. If...
Homework Statement
The following is Exercise 2.1 from from Ray d'Inverno's 'Introducing Einstein's Relativity.'
(a) Write down the Galilean transformation from observer ##S## to observer ##S'##, where ##S'## has velocity ##v_1## relative to ##S##.
(b) Find the transformation from ##S'## to...
Hi,
is there any general formula to find out the final velocity w, happened by a boost in x direction forst and then to y direction?
I could find the boost matrices for both and I know it's not purely another boost, rather a boost and a rotation, but I am really confused which formula to use...
Hello people,
I have a question regarding the x' component in the Lorentz/Galilean transformation.
So from what i understand is that there are 2 coordinate systems used in the transformations. One is used as a reference point and one is used for moving away from this point. The moving away in...