What is Lorentz boost: Definition and 41 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.
Before boost we have
Then using the Lorentz boost:
I want to calculate:
I tried multiplying the matrices together but I never get the stated answer which should be:
Using above formula, I could calculate the given commutator.
$$
[\epsilon^{\mu\nu\rho\sigma} M_{\mu \nu}M_{\rho\sigma},M_{\alpha\beta}]=i\epsilon^{\mu\nu\rho\sigma}(M_{\mu \nu}[M_{\rho\sigma},M_{\alpha\beta}]+[M_{\rho\sigma},M_{\alpha\beta}]M_{\mu \nu})
$$
(because...
Question:
Eq. 12.109:
My solution:
We’ll first use the configuration from figure 12.35 in the book Griffiths. Where the only difference is
that v_0 is in the z-direction. The electric field in the y-direction will be the same.
$$E_y = \frac{\sigma}{\epsilon _0}$$
Now we're going to derive the...
Pseudo-Riemannian manifolds (such as spacetime) are locally Minkowskian and this is very important for relativity since even in a highly curved spacetime, one could locally approximate the spacetime into a flat minkowski one.
However, this would be an approximation. Perhaps this is a naive...
I was reading a discussion where some physicists participated* where the topic of Lorentz invariance violations occurring in cosmology is mentioned.
There, they mention that we can imagine a Lorentz-violating solution to the cosmological equations. What do they mean by that? Can anyone specify...
Let's say we have some observer in some curved spacetime, and we have another observer moving relative to them with some velocity ##v## that is a significant fraction of ##c##. How would coordinates in this curved spacetime change between the two reference frames?
For example, imagine a...
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...
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##...
Let's begin with the first point.
a.I) Apply a generic boost in the y-z plane (take advantage of the arbitrariness in deciding the alignment of the y and z axes).
\begin{equation*}
B_{yz} =
\begin{pmatrix}
\gamma & 0 & -\gamma v_y & -\gamma v_z \\
0 & 1 & 0 & 0 \\
-\gamma v_y & 0 &...
Ateempt of solution:
There are two key coordinates in this scenario, the leftmost tip of the rod, which in ##S'## is ##C_{0} = (t', 0, ut',0)## and the opposite tip
##C_{1} = (t', L,ut',0)##
An angle ##\phi## could be found through a relationship such as ##tan(\phi) = \frac{ \Delta x}{ \Delta...
The Wikipedia article on Lorentz transformations is a bit confusing by its using speed and velocity almost interchangeably: of course γ (Gamma) stays the same, but (letting c=1) t'=γ(t-vx) , then if this is v⋅x, and x stays the same, then there would be a difference if something were going away...
Recently, I've been studying about Lorentz boosts and found out that two perpendicular Lorentz boosts equal to a rotation after a boost. Below is an example matrix multiplication of this happening:
$$
\left(
\begin{array}{cccc}
\frac{2}{\sqrt{3}} & 0 & -\frac{1}{\sqrt{3}} & 0 \\
0 & 1 & 0 & 0...
T = (x+\frac{1}{\alpha}) sinh(\alpha t)
X = (x+\frac{1}{\alpha}) cosh(\alpha t) - \frac{1}{\alpha}
Objective is to show that
ds^2 = -(1 +\alpha x)^2 dt^2 + dx^2
via finding dT and dX and inserting them into ds^2 = -dT^2 + dX^2
Incorrect attempt #1:
dT= (dx+\frac{1}{\alpha})...
I would like to apply a General Lorentz Boost to some Multi-partite Quantum State.
I have read several papers (like this) on the theory of boosting quantum states, but I have a hard time applying this theory to concrete examples.
Let us take a ##|\Phi^+\rangle## Bell State as an example, and...
What about if the speed parameter in a Lorentz boost were in fact related nontrivially to a Galilean speed ?
More formally ##L(v_L)=G(v)\circ F## where L is a Lorentz boost with Lorentz speed ##v_L##, G is a Galileo transformation with speed ##v## and ##F## is still an unknown linear...
In physics, a symmetry of the physical system is always associated with some conserved quantity.
That physical laws are invariant under the observer’s displacement in position leads to conservation of momentum.
Invariance under rotation leads to conservation of angular momentum, and under...
Homework Statement
I am reading through Griffiths' Electrodynamics, and I have come to the scenario in the Relativity chapter where in an inertial reference frame ##S##, we have a wire, with positive charges (linear density ##\lambda##) moving to the right at speed ##v##, and negative charges...
Homework Statement
Given an electromagnetic tensor ##F^{\mu\nu}##, showing that:
$$\det{F^{\mu}}_\nu=-(\vec{B}\cdot\vec{E})^2$$
Homework Equations
The Attempt at a Solution
I had only the (stupid) idea of writing explictly the matrix associated with the electromagnetic tensor and calculating...
Homework Statement
In a reference frame ##S## there is a particle with mass ##m## and charge ##q## which is moving with velocity ##\vec{u}## in an electric field ##\vec{E}## and in a magnetic field ##\vec{B}##. Knowing the relativisitc laws of motion for a particle in an EM field, find the...
I've been working my way through Peskin and Schroeder and am currently on the sub-section about how spinors transform under Lorentz transformation. As I understand it, under a Lorentz transformation, a spinor ##\psi## transforms as $$\psi\rightarrow S(\Lambda)\psi$$ where...
Compare this with the definition of the inverse transformation Λ-1:
Λ-1Λ = I or (Λ−1)ανΛνβ = δαβ,...(1.33)
where I is the 4×4 indentity matrix. The indexes of Λ−1 are superscript for the first and subscript for the second as before, and the matrix product is formed as usual by summing over...
Hi! I came out with a problem last night I wasn't able to solve:
Let's assume we have a condensator with a uniform electric field E confined in its inside, lying on the z axes. Let's also assume we have a piece of a ferromagnetic object aligned with the condensator at time t = 0, on the y-axes...
Homework Statement
Suppose given an electric field \vec{E} and a magnetic field \vec{B} in some inertial frame. Determine the conditions under which there exists a Lorentz transformation to another inertial frame in which \vec{E} || \vec{B}
Homework Equations
If we give a Lorentz boost along...
Hi. First, excuse my English.
In my lecture notes on classical electrodynamics, we are introduced to the Lorentz transformations: a system S' moves relative to a system S with positive veloticy v in the x-axis (meassured in S), spatial axis are parallel, origin of times t and t' coincide...
The Schwarzschild Metric (with ##c=1##),
$$ds^2 = -\Big(1-\frac{2GM}{r}\Big)dt^2+\Big(1-\frac{2GM}{r}\Big)^{-1}dr^2+r^2d\Omega^2$$
can be adjusted to a form involving three rectangular coordinates ##x##, ##y##, and ##z##:
$$ds^2 =...
I'm trying to understand the relativistically spinning disk within the framework of SR (if that is even possible). I thought to first simplify the problem by considering a spinning ring/annulus, but I don't know if my analysis is correct.
I imagined a spinning ring of radius R, spinning at an...
Hello,
If I have a momenta pμ=(E,px,py,pz) and transform it via lorentz boost in x-direction with velocity v I'll get for the new 0th component E′=γE+γvpx why is this in the limit of low velocities the same as transforming the energy by a galilei transformation with velocity v? For γvpx i get...
Homework Statement
In the inertial frame of observer A two events occur at the same position a time 10 ns apart. In the frame of the observer B moving with respect to RA, one event occurs 1m away from the other. What is the difference in time between the two events in B's frame.
Solve by...
As I understand it, the value of a 4-vector x in another reference frame (x') with the same orientation can be derived using the Lorentz boost matrix \bf{\lambda} by x'=\lambda x. More explicitly,
$$\begin{bmatrix}
x'_0\\
x'_1\\
x'_2\\
x'_3\\
\end{bmatrix}
=
\begin{bmatrix}...
Hello!
I'm trying to derive the general matrix form of a lorentz boost by using the generators of rotations and boosts:
I already managed to get the matrices that represent boosts in the direction of one axis, but when trying to combine them to get a boost in an arbitrary direction I always...
On page 29 equations 2.1.20 and 2.1.21 of Gravitation and Cosmology by S. Weinberg he gives these expresions for matrix componentes:
\Lambda_j^0=\gamma v_j
My question is: shouldn't there be a minus sign on left side of the equation?
Greetings,
I have been having trouble deriving the equation for the general Lorentz boost for velocity in an arbitrary direction. It seems to me that given the 1D Lorentz transformations...
matrix for Lorentz transformation in x-direction, X:
{{1/sqrt(1-v^2), -v/sqrt(1-v^2), 0, 0}...
Homework Statement
i) Show that the wave equation:
[( -1/c^2) d^2/dt^2 + d^2/dx^2 + d^2/dy^2 + d^2/dz^2 ]u(t,x,y,z) = 0
is invariant under a Lorentz boost along the x-direction, i.e. it takes the same form as a partial differential equation in the new coordinates. [Use the chain rule in two...
Homework Statement
Given that (φ/c,A) is a 4-vector, show that the electric field component Ex for a
Lorentz boost along the x-axis transforms according to Ex' = Ex.
Homework Equations
E_x = -\frac{\partial \phi}{\partial x} - \frac{\partial A_x}{\partial t}
A_x being the x component of the...
Hi i have a problem with some work.
a muon type neutrino interacts with a stationary electron, producing a muon and electron type neutrino. I have calculated the CM energy but now need to calculate gamma, the lorentz boost.
γ=(Eν/2me)^1/2
How do i show this? the info i have is that...
Homework Statement
So, I'm working through a relativity book and I'm having trouble deriving the Lorentz transformation for an arbitrary direction v=(v_{x},v_{y},v_{z}):
\[\begin{pmatrix}
{ct}'\\
{x}'\\
{y}'\\
{z}'
\end{pmatrix}=\begin{pmatrix}
\gamma & -\gamma \beta _{x} &...
I was thinking that if i have for example a boost in the direction of x, then the boost should be equivalent to an hyperbolic rotation of the y and z axes in the other direction. I don't know if it's true or not. Then I want to know if somebody knows this result or why is false?
I was...
I was reading a section on lorentz boosts and i need some help understanding what they did:
the book starts off by defining the line element dS where:
(dS)^2 = -(CΔt)^2 + dx^2 + dy^2 + dz^2
then they say: "consider the analogs of rotations in the (ct) plane. These transformations leave...
hi there!
Just wondering... if i have a photon moving in the z direction 4 momentum given by (0,0,1,1)
and I lorentz boost it in the z direction... would I get the same original 4 momentum (0,0,1,1) because i thought that boosting something at the speed of light means that it remains at...
Einstein Summation Convention / Lorentz "Boost"
Homework Statement
I'm struggling to understand the Einstein Summation Convention - it's the first time I've used it. Would someone be able to explain it in the following context?
Lorentz transformations and rotations can be expressed in...