# 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.

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1. ### Problem Related to Photons with Mass

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:
2. ### Lorentz boost generator commutator

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...
3. ### Field transformations in the z-direction

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...
4. ### I Approximate local flatness = Approximate local symmetries?

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5. ### I Solutions that break the Lorentz invariance...?

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...
6. ### I Gravitational Field Transformations Under Boosted Velocity

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...
7. ### I Only Minkowski or Galilei from Commutative Velocity Composition

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8. ### I Deriving Lorentz Transformations: Hyperbolic Functions

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##...
9. ### Calculate a specific boost and rotation

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10. ### Moving rod seems inclined

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11. ### I Lorentz boost -- speed or velocity?

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...

27. ### I Understanding Relativistically Spinning Disk/Ring: Lorentz Boosts

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...
28. ### Relativistic momentum (Lorentz boost) low velocity limit

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...
29. ### Finding Lorentz boost speed

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...
30. ### Lorentz boost matrix in terms of four-velocity

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}...
31. ### General matrix representation of lorentz boost

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...
32. ### Confusion with Lorentz boost

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?
33. ### Deriving General Lorentz Boost 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}...
34. ### Performing a lorentz boost

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35. ### Lorentz boost, electric field along x-axis, maths confusion?

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...
36. ### The lorentz boost of the CM frame w/ respect to the lab frame

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...
37. ### Deriving the Lorentz Boost for an Arbitrary Direction

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} &...
38. ### Lorentz boost and equivalence with 3d hyperbolic rotations

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...
39. ### Lorentz Boost Help: Why Use Hyperbolic Functions?

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...
40. ### Lorentz Boost of a photon

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41. ### Einstein Summation Convention / Lorentz Boost

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