Lorentz transformation Definition and 380 Threads

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. A

    Show EM Wave equation invariant under a Lorentz Transformation

    Homework Statement Show that the electromagnetic wave equation \frac{\partial^{2}\phi}{\partial x^{2}} + \frac{\partial^{2}\phi}{\partial y^{2}} + \frac{\partial^{2}\phi}{\partial z^{2}} - \frac{1}{c^2}\frac{\partial^{2} \phi}{\partial t^2} is invariant under a Lorentz transformation...
  2. C

    Lorentz transformation where electric field vanishes

    Homework Statement We have an homogeneus electromagnetic field with E \bullet B=0 and E \neq cB Find the velocity of the reference frames in which ony E exists. Homework Equations \mathbf{E}' = \gamma \left( \mathbf{E} + \mathbf{v} \times \mathbf{B} \right ) - \left...
  3. S

    Lorentz Transformation Limit: Proving U=c

    G'day, I'm just doing some physics homework and decided to attempt to prove something. This is not a homework problem, I'm just unsure how to evaluate the limit. Using the equation for transformation of velocity U=(U'+V)/(1+(VU'/c2)), I'm trying to show that if V=-c, as U' approaches c, U...
  4. N

    Question about the Lorentz Transformation

    http://img710.imageshack.us/img710/4081/capturertt.jpg http://img233.imageshack.us/img233/9885/capturejpg2f.jpg [PLAIN]http://img52.imageshack.us/img52/9140/capturejpg3.jpg [PLAIN][PLAIN]http://img813.imageshack.us/img813/2919/capturejpg4.jpg...
  5. L

    Algebra help with Lorentz Transformation

    Homework Statement Hey all, my algebra isn't as great as it used to be and I am having trouble with some of the algebra dealing with Lorentz transformation. Basically, I just need someone to do a quick step by step of how to go from x' to x (see the following). I started with the top...
  6. D

    Lorentz Transformation Applicability Re: EM & Casuality

    Hi, From what I've learned so far, Lorentz transformation meets certain criteria, such as the constancy of EM wave propagation speed in vacuum, &/ casuality, among others. My question is, why would it/would it not be applicable to phenomena that have nothing to do with EM interaction? In...
  7. M

    Lorentz transformation, special relativity problem

    Homework Statement Frame S and S' are moving with respect to each other in the x-axis with some velocity. An event happens in S' at x'_1 = 1.0 c*year at t'_1 = 1.0 year. Another event happens at t'_2 = 2.0 c*year at t'_2 = 0.5 year. The two events are simultaneous at some point in S. The...
  8. B

    Special relativity: Simple Lorentz transformation question

    Homework Statement Observer O sees a fire-engine leave its station 6363 m due north from Cape Canaveral, where the super-shuttle Lorentz had been launched 10^-5 s earlier. A space-cruiser flying north-east sees these two events also 10^-5 s apart, but with the shuttle launch occurring after...
  9. S

    Deriving the Lorentz transformation

    Homework Statement Derive the Lorentz transformation by assuming that the transformation is linear, and does not change the perpendicular coordinates. Write the transformation as x' = A1 (x - vt), y' = y, z' = z, t' = A2 t + A3 x, Determine A1, A2, A3 by requiring that a flash of light...
  10. M

    Relativity - Lorentz Transformation

    Homework Statement A is at the base station and given in K co-ordinates B is on a spacecraft and given in K' co-ordinates. The velocity of the spacecraft is v=0.8c Question 1 After t = 2y (y = years) A sends a message by radio to B demanding a picture. Which time t' does B have when...
  11. Z

    Lorentz Transformation - Proof that t'2 - t'1 >0

    A 'cause' occurs at point 1 (x1, t1) and its 'effect' occurs at point 2 (x2, t2) as measured by observer O. Use Lorentz transformation to find t'2 - t'1 as measured by O' and show that t'2 - t'1 >= 0. that is Observer O' can never see the effect before the cause. I know that is possible to...
  12. alemsalem

    What happens to electron spin under Lorentz transformation?

    When you start in the rest frame of the electron, the Spinor w(p = 0) = (1 0 0 0 ) represents a positive energy state with spin up in the Z direction u = (0, 0 0 1),, that is the spinor is an eigenspinor of the operator S . u, where S is a 4 dimensional operator (S0, S) after a Lorentz...
  13. P

    Length Contraction: Need Help Understanding Lorentz Transformation

    Hello! need some help with length contraction. So according to lorentz transformation we got I don't know how to put symbols so ill use Y as gamma since they look alike :) dx' = Y dx - u Y dt So proper length refers to the frame where dt = 0 since u are measuring the ends at the...
  14. G

    Improper Lorentz transformation

    hey, I heard about improper Lorentz transformations, but I did not really understand what they are. I just know that their determinant is -1, but what does this physically mean? Can somebody explain this to me?
  15. S

    Lorentz transformation and rockets

    Homework Statement 1The rockets of the Goths and the Huns are each 1000 m long in their rest frames. The rockets pass each other, virtually touching, at a relative speed of 0.8c. The Huns have a laser cannon at the rear of the rocket that shoots a deadly laser beam at right angles to the...
  16. D

    Lorentz Transformation: Exploring Special Relativity

    Hello This is a part of a simple paper about special relativity [PLAIN]http://img15.imageshack.us/img15/8789/91001769.jpg I don't understand the assumption in the red box .. why are they all squared ? thank you
  17. A

    If correct: a catastrophe in the Lorentz transformation

    The Lorentz transformation are given by (see the attachment) x'=(x-vt)/√(1-v^2/c^2 ) t'=(t-vx/c^2)/√(1-v^2/c^2 ) Let us transform the event (10^100 m,1sec) in the x-frame to the x'-frame that is moving in the usual geometry with the speed v=10^(-10) c. Could you see that that t'≈-10^81...
  18. G

    Lorentz transformation in Coulomb gauge

    Hello I have been having trouble understanding equation 14.25 in Bjorken and Drell "Relativistic Quantum Fields" and how exactly it gets to it. Also I would like to explicitly find/derive what the operator gauge function is. Can anyone help please?
  19. E

    Commuting the Lorentz transformation with derivative

    In the process of proving that the d'Alembert operator https://www.physicsforums.com/attachments/31306 is invariant under Lorentz transformations, it was required to commute two terms in the following expression for the transformed operator, which was obtained by switching the index on the...
  20. P

    Length Contraction of Particles & Photons in Relativity

    I'm trying to teach myself special relativity. I use the book 'Introduction to Special Relativity' by Wolfgang Rindler. I have a question about length contraction. We consider 2 particles traveling along the x-axis of a reference frame S with a constant distance between them. We can always go...
  21. P

    Lorentz Transformation Question

    Homework Statement 2 particles are created in a high-energy accelerator and move off in opposite directions. The speed of one particle, as measured in the laboratory is 0.650c, and the speed of each particle relative to the other is 0.950c. What is the speed of the second particle, as measured...
  22. L

    D'alembertian of lorentz transformation matrix

    Is the d'alembertian of lorentz transformation matrix 0? and why? would it be 0 because it lorentz invariant? thanks
  23. H

    Trying to understand the Lorentz transformation.

    Ok so I am attempting to get a "feel" of the Lorentz equations. For a observer O' moving with velocity v respect to a observer O along the x direction the transformed variables are x and t. x' = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}(x - vt) t' = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}(t -...
  24. P

    Exploring Lorentz Transformation Matrix: A'^{\mu}=\alpha^{\mu}_{\nu}A^{\nu}

    \alpha=\left(\begin{array}{cccc} \gamma& 0&0& -\beta\gamma\\ 0&1& 0 & 0\\ 0 & 0 & 1 & 0\\ -\beta\gamma & 0 & 0 & \gamma \end{array} \right)x'^{\mu}=\alpha^{\mu}_{\nu} x^{\nu} \alpha is Lorrentz transformation matrix. Can I see something more about it? . It's symmetric. That is important...
  25. P

    Lorentz Transformation: Coefficients a_{nm}(u) & Inverse Relations

    x'=a_{11}x+a_{12}y+a_{13}z+a_{14}t y'=a_{21}x+a_{22}y+a_{23}z+a_{24}t z'=a_{31}x+a_{32}y+a_{33}z+a_{34}t t'=a_{41}x+a_{42}y+a_{43}z+a_{44}t \vec{u}=u\vec{e}_x Coefficients a_{nm}=a_{nm}(u) Why I suppose that coefficients are function only of velocity u? Inverse relations...
  26. B

    How Does Rapidity Influence the Lorentz Transformation Matrix?

    1. Homework Statement : Consider a two dimensional Minkowski space (1 spatial, 1 time dimension). What is the condition on a transformation matrix \Lambda, such that the inner product is preserved? Solve this condition in terms of the rapidity. 2. Homework Equations : Rapidity Relations...
  27. T

    Srednicki 2.8 / Inverse Lorentz Transformation

    Homework Statement I need to demonstrate the relation [\varphi(x),M^{\mu\nu}]=\matchal{L}^{\mu\nu}\varphi(x) where \mathcal{L}^{\mu\nu}\equiv \frac{\hbar}{i}(x^\mu\partial^\nu-x^\nu\partial^\mu). Homework Equations U(\Lambda)^{-1}\varphi(x)U(\Lambda) = \varphi(\Lambda^{-1}x) \Lambda =...
  28. D

    Question on lorentz transformation equations

    i am reading Lillian R. Lieber's book on the einstein theory of relativity and i am a bit confused on page 65. she wants to take the equations: x=x'cosθ - y'sinθ y=x'sinθ + y'cosθ and compare them to: x'=β(x-vt) t'=β(t-vx/c2) she takes c as one so: x'=β(x-vt) t'=β(t-vx) she...
  29. T

    Help with Lorentz Transformation

    In Einstein's book Relativity he provides a derivation of the LT. link here http://www.bartleby.com/173/a1.html" In step 3 he brings in constants λ and μ and now I am lost. In the equation (x'-ct') = λ(x-ct) - isn't this the same as "zero = anything X zero"? How did λ and μ get...
  30. Y

    Lorentz transformation of friction

    In special relativity, velocity dependent forces transform. Let us then consider frictional forces, such as drag, which are velocity dependent in the first order. Do two observers moving relative to a third body measure different frictional effects?
  31. K

    Calculating Muon Velocity with Lorentz Transformation | Physics Homework

    Homework Statement Muons, which have a half-life of 2 x 10-6 s, are formed in the Earth's atmosphere at an altitude of 10 km. If they travel normal to the Earth's surface, and one half of them reach it before they decay, what is their velocity? Homework Equations Lorentz...
  32. C

    How Does the Lorentz Transformation Affect Observed Velocity?

    If an observer O' see a body that moves with constant velocity Ux along the axis x' in a positive direction. What is the velocity VxO of body observed by O? That is all the information I have. Can you help? :blushing:
  33. K

    T: Lorentz Transformation for Length and Relativistic Mass

    Using the Lorentz transformation, at what speed relative to speed of light c must you travel so that your length along the direction of motion is seen to decrease by a factor of 2? For this speed, hwo much would your mass be increased? Please show steps, I'm confused with this question!
  34. C

    Lorentz transformation as a higher dimensional rotation

    From this link http://en.wikipedia.org/wiki/Introduction_to_special_relativity" In the section entitled "Invariance of Length: The Euclidean Picture" the article discusses how rotations within an n-dimensional space keep length invariant. However, if you rotate and object into a higher, n+1...
  35. N

    How Do You Calculate Velocity in Lorentz Transformations?

    I'm studying for my modern physics final and this problem is giving me trouble; Q: In a frame S, two events have spatial separation deltaX= 600m, delta y and delta z = 0, and a temporal separation deltaT= 1micro second. A second frame S' is moving along the same axis with nonzero speed v...
  36. facenian

    Lorentz Transformation: R_2=R_1^{-1}?

    Let L_w be a Lorentz transformation between to systems that coincide at t=0(paralell axes assumed) and with relative velocity w along x_1. If L_u is the Lorentz transformation when the relative velocity u is in any direcction then we have that L_u=R_2 L_w R_1 where R_2 and R_1 are sapce...
  37. B

    Simple Lorentz transformation problem

    Homework Statement 3. At what speed (measured in units so c = 1) must the train be moving in order for the points(X,T) = (1,1) and (X,T) = (5,2) to be simultaneous in the (X',T') coordinate system? Homework Equations Disclaimer: I'm not actually in a physics class, I'm in an elementary...
  38. S

    How Do You Calculate the Inverse Lorentz Transformation Matrix?

    Homework Statement Let \Lambda^{\bar{\alpha}}_{\beta} be the matrix of the Lorentz transformation from O to \bar{O} , given as: \bar{t} = \frac{t-vx}{\sqrt{1-v^2}}, \bar{x} = \frac{-vt+x}{\sqrt{1-v^2}}, \bar{z} = z, \bar {y} = y . Let \vec{A} be an arbitrary vector with components...
  39. V

    Lorentz transformation in four dimensions of the electromagnetic tensor

    Homework Statement Given: electromagnetic tensor F(superscript)uv: electromagnetic tensor F' after the lorentz transformation: [ 0 -Ex -gamma(Ey-VBz/c) -gamma(Ez-VBy/c) Ex 0...
  40. G

    Lorentz transformation using to find for vx of a missile

    I have a question which i believe is pretty standard, spaceship traveling towards space staion at 0.80c it fires a missile at 0.40c what is the speed of the missle observed by the space station I'll tell you where I am at; I have set 2 inertial frames. A being that of the space station and...
  41. P

    Modern Physics - Extention of the Lorentz Transformation?

    Homework Statement Conventionally, the Lorentz Transformation relates two reference frames that begin at the same location and time with one reference frame moving at a constant velocity {\vec{v}} along a positive {x}-axis (which is common to both reference frames) with respect to the other...
  42. J

    Lorentz Transformation: Velocity Transformation vs Addition

    Is there a difference between the Lorentz velocity transformation and the relativistic velocity addition? They give the same result...
  43. E

    Lorentz transformation: time dilation

    1.) problem statement Relativistic protons that have a certain speed "v" are selected by measuring the time it takes the proton to travel between two detectors separated by a distance "L". Each detector produces an electronic pulse of very short duration (LaTeX Code: \\Delta t << L/v) when a...
  44. N

    What is the Distance Between Two Photons in Different Reference Frames?

    Homework Statement two photons travel along the x-axis of S , WITH A CONSTANT DISTANCE L betweenthem. Prove that in S's the distance between these photons is L(c+v)^1/2/(c-v)^1/2. Homework Equations x'=gamma*(x-vt), x=gamma*(x'+vt), t=gamma*(t'+vx'/c^2), t=gamma*(t'-vx'/c^2) The...
  45. K

    Basis vectors under a Lorentz transformation

    Hello, I am new to the forums and I hope this fundamental topic has not been previously treated, as these forums don't seem to have a search function. I am studying general relativity using S. Carroll's book (Geometry and Spacetime) and I am having a fundamental problem with basis vectors under...
  46. E

    Lorentz Transformation (simplified)

    We know that many books have deduced Lorentz transformations through rigorous maths and they add little to our visions about what's going on. But in the pdf I have attached, I have tried to deduce this transformation with logical arguments. It is really simple and no tensors have been included...
  47. N

    Is the Energy Pulse Responsible for Planet X's Explosion?

    "a spaceship goes from Earth to planet x. then it goes to the moon of planet x.when it reaches the moon it detects a energy pulse and 1.01 seconds later planet x blows up. the distance between planet x and its moon is 400 million meters the speed of the spaceship relatively to planet x and its...
  48. Y

    Galileo and Lorentz transformation

    Though I believe I have understood some basic ideas, theories and mathematic formulas of SR, I still have a pretty fundamental question: Many textbooks start SR with a light clock consisting of two mirrors and a light blip bouncing in between, claiming that when the light clock moves, the...
  49. bcrowell

    Simple argument for area conservation under Lorentz transformation?

    A Lorentz boost along the x-axis conserves area in the x-t plane. Can anyone think of an argument to prove this fact that's simpler than the one below? I don't want to assume the form of the Lorentz transformation, because I want to use the conservation of area as part of a derivation of the...
  50. E

    Lorentz Transformation Vector Form Formula

    What is the vector form formula (not the martrice form) for lorentz transformation from system S' to S for a particle with 4-vectors (\vec{r'},ct') and (\vec{p'},E'/c) where S moves with an arbitary speed \vec{v} in relation to S'? thanks
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