What is Lorentz transform: Definition and 57 Discussions
In physics, the Lorentz transformations are a sixparameter 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 xdirection, 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(xvt\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 xaxis, 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 noninertial (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 rotationfree 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.
Here is the spacetime diagram of an observer:
Here is the diagram as seen by an observer travelling from left to right:
I have attempted to represent the axis system of the moving observer on the axis system of the stationary observer in the following diagram:
Event D seems to lie in...
I have a quick question about the Galilean transform. If I have Alice running and Bob stationary. The Galilean transform will tell me the position of Alice from Bob's stationary position. Also if I have Alice's position which is moving it will tell me Bob's stationary position.
If I want Bob...
Image below. Is the Lorentz transform just switching between a stationary frame and a moving frame?
I forgot to write Alice's frame but I assume that is obvious.
I read in one book about the deduction of Lorentz transform. It writes:
'
$$
\begin{aligned}
t^\prime & = \xi t + \zeta x (1) \\
x^\prime & = \gamma x + \delta t (2) \\
y^\prime & = y (3) \\
z^\prime & = z (4)
\end{aligned}
$$
from (2), it gives:
$$
\begin{aligned}
{dx \over dt} = { \delta...
Obviously in the title I mention the user that recently got banned, but the reason I do is because s/he was having some trouble accepting that a B field transforms into a mixture of E and B fields per the Lorentz transformation (and other assorted quackery), so it got me thinking about why this...
Good afternoon,
Not sure if this should be in the homework section or not but in any case...
I'm having difficulty understanding the outputs from the Lorentz transform.
Example problem.
The Earth and sun are 8.3 lightminutes apart. Ignore their relative motion for this problem and assume...
Hi all, just had a question about tensor/matrix notation with the inverse Lorentz transform. The topic was covered well here, but I’m still having trouble relating this with an equation in Schutz Intro to GR...
So I can use the following to get an equation for the inverse...
Homework Statement
i)A police spaceship P is chasing another spaceship A. Both ships have velocities βP = βA = 3c/5 as measured along the xaxis in the Solar System reference frame O. The police ship is a distance L = 1 lightsecond (i.e. the distance traveled by light in one second) behind...
Let ##\Lambda## be a Lorentz transformation. The matrix representing the Lorentz transformation is written as ##\Lambda^\mu{}_\nu##, the first index referring to the rows and the second index referring to columns.
The defining relation (necessary and sufficient) for Lorentz transforms is...
The Lorentz transformation operator acting on an undotted, i.e. righthanded, spinor can be expressed as $$e^{\frac{1}{2} \sigma \cdot \mathbf{\phi} + i\frac{1}{2} \sigma \cdot \mathbf{\theta}}.$$
There is a very cool, almost childlike, derivation of this expression in Landau Vol. 4 S. 18 I've...
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...
I was reading this paper: http://dinamico2.unibg.it/recami/erasmo%20docs/SomeOld/RevisitingSLTsLNC1982.pdf
It is on superluminal Lorentz transformations and is too advanced for me. :confused:
But anyway, take a look at equation(s) (11). For the y' and z' transformations, there is an imaginary...
Alonso has in fact raised a relevant issue. Given any clock synchronization in the embankment frame, there is a unique pair of points at the embankment with coordinates x=a and x=b, and a unique time t=T, such that at t=T in the embankment frame, these points are next to the back end and the...
Reading Griffiths, he states that the Lorentz Transform is useful for describing where an 'event' occurs in a different inertial frame. What about describing the motion of a particle in this moving frame if I know its motion in my frame?
Really, I'm looking at pickup ions in the solar wind. A...
When one considers a Lorentz transformation between two frames ##S## and ##S'##, such that the coordinates in ##S## are given by ##x^{\mu}## and the coordinates in ##S'## are given by ##x'^{\mu}##, with the two related by x'^{\mu}=\Lambda^{\mu}_{\;\;\nu}x^{\nu} then a scalar field ##\phi (x)##...
I'm trying to prove the following statements relating to spacelike, timelike and lightlike spacetime intervals:
1. There exists a reference frame in which two spacetime events are simultaneous if and only if the two events are spacelike separated.
2. There exists a reference frame in...
Do two inertial observers in relative motion agree on their relative velocity? Velocity is distance per unit time and they don’t agree on the distance or the elapsed time. If the apparent distance in the prime system is shorter and the elapsed time is longer, then it seems that the apparent...
Prof Ramamurti Shankar has this Youtube video 'Introduction to Relativity' at
And in it he derives the Lorentz Transforms something like this, at about 58 minutes into it.
 t  time
ut x' ...
Hi all,I am trying to understand relativity and Lorentz Transformation more clearly but I have some problems. Assume that we have frame F' which is moving at velocity v with respect to F. Now assume we have an object, O, moving at velocity, w, with respect to F. Frame F has its own time, t, and...
Homework Statement
Given that ##x_\mu x^\mu = y_\mu y^\mu## under a Lorentz transform (##x^\mu \rightarrow y^\mu##, ##x_\mu \rightarrow y_\mu##), and that ##x^\mu \rightarrow y^\mu = \Lambda^\mu{}_\nu x^\nu##, show that ##x_\mu \rightarrow y_\mu = \Lambda_\mu{}^\nu x_\nu##.
Homework Equations...
Homework Statement
A traveler in a rocket of length 2d sets up a coordinate system S' with origin O' anchored at the exact middle of the rocket and the x' axis along the rocket’s length. At t' = 0 she ignites a flashbulb at O'. (a) Write down the coordinates t'_F, x'_F, and t'_B, x'_B for...
A first stage of the determination.
We have a body of length L = AB, which moves along the xaxis with velocity v,
say that coming to us.
A  B v <

 h  vertical distance
 ./  a light converges to us with the speed eq. c
 /
O > x
At some time t = 0, we see the point A...
Basically I am trying to lorentz transform the magnetic field along θ of a bunch particles which have a gaussian distribution to the radial electric field. However the magnetic field in θ is dependent on the longitiudinal distribution.
Now initially i thought we would just use the standard LT...
We have a few posters struggling with this, I thought I'd post a step by step guide, to see if it would help. That seems easier than trying to untangle the confused threads we have. We'll see if it works...
Setup and notation:
We have a rocket, which has a front and a back.
We have a...
Homework Statement
The problem can be found in Jackson's book.
An infinitesimal Lorentz transform and its inverse can be written under the form ##x^{'\alpha}=(\eta ^{\alpha \beta}+\epsilon ^{\alpha \beta})x_{\beta}## and ##x^\alpha = (\eta ^{\alpha \beta}+\epsilon ^{'\alpha \beta})...
Homework Statement
I must show that the one dimensional wave equation ##\frac{1}{c^2} \frac{\partial u}{\partial t^2}\frac{\partial ^2 u}{\partial x^2}=0## is invariant under the Lorentz transformation ##t'=\gamma \left ( t\frac{xv}{c^2} \right )## , ##x'=\gamma (xvt)##Homework Equations...
Why do we say that t'=t for Galilean transformation, when the low velocity limit of the Lorentz transformation is t'=t+vx/c2?
If x is really big, then doesn't time cease to be absolute, no matter how small v/c is?
Hi!
I am new here, thought to join as I am trying to learn Relativity, in this case Special Relativity. I have solved a bunch of problems already but ...
The Lorentz Transform formulation I am dealing with is a 4x4 matrix. I understand the invariance of the spacetime interval and have...
Homework Statement
Derive the relation:
E' = \gamma (E + \beta p c)Homework Equations
p' = \gamma p \\
E^{2} = p^{2} c^{2} + M^{2}c^{4}The Attempt at a Solution
start off with stationary frame S E=mc^{2}
then in moving frame S' E'^{2} = p'^{2} c^{2} + E^{2}:
lorent transform momentum:
E'^{2} =...
I have recently found that Mathematica has a "reader" which allows interactive calculations and plots to be shared. Here is a simple example, please let me know if there is interest in more such widgets.
Homework Statement
Consider an infinite line of charge with density λ per unit length lying along the zaxis. If the line of charge is stationary in frame S, use the Lorentz transform to find the current and charge densities in a frame S' which is moving with velocity v parallel to the...
OK, I've found a great explanation of the derivation of the Lorentz transformation, with
x' = γ [ x  v t ]
t' = γ [ t  ( v / c2 ) x ]
so if I take the other term as 0, there is
x'( t = 0 ) = γ x
t'( x = 0 ) = γ t
but the problem is that the time dilation & length...
If you Lorentz transform a scalar:
U^{1}(\Lambda)\phi(x)U(\Lambda)=\phi(\Lambda^{1}x)
If you now perform another Lorentz transform, would it it look like this:
U^{1}(\Lambda')U^{1}(\Lambda)\phi(x)U(\Lambda)U(\Lambda')=\phi(\Lambda'^{1}\Lambda^{1}x) ?
But isn't this wrong...
I'm trying to follow along with Simple Derivation of the Lorentz Transformation, but am having some hurdles.
I'll be referring to step (5) which states:
x'=axbct
ct'=actbx
In paragraph marked 6, I see that the author tries to get eqn (5) to describe motion of the K' frame. This is an...
Homework Statement
see attached .pdf. all parts of problem statement are italicized.
Homework Equations
see attached .pdf
The Attempt at a Solution
see attached .pdf
Actually: my question is pretty qualitative. You can look at everything I've done with this problem so far...
Homework Statement
From Lorentz Transform,
x^{\prime} = \gamma (x  vt)
From textbooks and wikipedia,
L_0 = x'_2  x'_1 = \gamma (x_2  x_1 )
Where x_1 and x_2 = L
Thus,
\L_0 = \gamma L
Question is this:
If i take the same method and us the Inverse Lorentz transform, i seem to...
Hey all,
Simple question, so hopefully it gets answered quickly.
I hate SR, but am marking an assignment on it. Here's the setup
Alice is standing still on a train traveling at 3/5c and Bob is on the platform at rest. A series of lights are set up along the track, yada yada. Bob sees...
Hi,
I'm reading a book on SR/Field theories that simply posits the spacetime interval and from that defines a Lorentz transform as any transformation which leaves the interval invariant. My question is how do we posit the spacetime interval in this manner using only the postulates of...
Homework Statement
Show that a Lorentz transformation preserves the sign of the energy of a solution to the Dirac equation.
The Attempt at a Solution
I'm not sure how to approach this.
So I apply the Lorentz transform to the Dirac equation, and work through it to obtain the energy...
Homework Statement
A spaceship of proper length Lp = 400 m moves past a transmitting station at a speed of v = 0.76c. At the instant that the nose of the spaceship passes the transmitter, clocks at the transmitter and in the nose of the spaceship are synchronized to t = t' = 0. The instant...
Hi..I was studying Ryder, Chapter 2[Quantum Field Theory]...he derives the Dirac eq using Lorentz transformations..I found the approach fascinating..but there is one part I do not really understand...
Just a few lines before he writes down the Dirac equation, he identifies \varphi_{R}(0) with...
Homework Statement
Derive the Lorentz transformation for the x component of momentum, i.e.
Px' = \gamma (Px  vE/(c^{}2))
I've used Px = x component of momentum (not very good with latex, sorry!)
Homework Equations
I thought the best place to start was the Lorentz transformation...
Homework Statement
Astronomers on the Earth (regarded as an inertial reference frame) see two novas flare up simultaneously. One of the novas is at a distance of 1.0x10^3 lightyears in the constelation Draco; the other nova is at an equal distance in the constellation Tucana in a direction (as...
If we impose that under no force or gravitational field Lorentz transform must hold, my question is if the validity of Lorentz transform means that space time must be continous, or on the other hand if there may be a discrete model of spacetime that preserves Lorentz transform.
For example...
What is the equivalent of the Lorentz Transform when the metric is not Minkowski? How do you do a coordinate transform with a metric that has nondiagonal terms?
Homework Statement
Suppose that a particle of mass m and energy E is moving toward the origin of a system S such that its velocity u makes an angle alpha with the yaxis (approaches origin from upper right). Using the Lorentz transformations for energy and momentum, determine the energy E' of...
URGENT* Using Lorentz transform to find velocity of rod
Homework Statement
a person at the origin of an inertial reference frame S observes a rod of proper length l moving towards him at a speed v. He notes that the rod takes a time T to pass him. Assuing that when the front end of the rod...