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### Course: Physics library>Unit 16

Lesson 3: Lorentz transformation

# Deriving Lorentz transformation part 2

Continuing the algebra to solve for the Lorentz factor.

## Want to join the conversation?

• But what if you take a point that isnt on the diagonal. (light). doesnt this assume that we are talking about light or something like that?
• The points on the diagonal world line of light is used to calculate the coordinate transformations required to keep the world line of light at a 45 degree angle.

Once the transform is derived you can use it for transforming the coordinates that are not one the diagonal world line of light.
• Aren't x=ct and x"=ct" only true when v=c?
• I'm confused; if we assumed that the event we were observing is a beam of light (or at least connected with light) shouldn't the transformations be only true in the case of light observation? And can we prove the Lorentz transformation assuming the observed object isn't connected with a light beam?
• Why do we multiply x and a prime? I know that it gives us the scaling factor but what is the significance of multiplying x and x prime? On what basis we multiply it?
• Because we say light has absolute speed and going with it everytime. This means light always has 45degree in minkowski diagrams. You can check pictures about this.

Also think like that, you shooted light and it moved for a sec, means 3x10^8meter, you check it on x side. Now look for ct side, t is 1 and c is 3x10^8 as you know. They same for anytime.
(1 vote)
• I still don't understand why the light path is where it is (always at a 45 degree angle). Please explain if you understand. Thank you.
(1 vote)
• That's how the axes of the graph are set up.. The unit on the axis is t and on the y axis it is ct. That means the slope of the graph is c, and if you draw equal-sized units, the line will be at 45 degrees because for every extra unit of time, light travels one extra unit of ct in distance.e
(1 vote)
• Why would we not substitute in c for v as well? Isn't v just the velocity of whatever is moving, and since it is light in this case, wouldn't that make v equal to c?
• Why does the scaling of the Galilean transformations by the factor Y=[1/sqrt(1-v^2/c^2)]give us the Lorentz transformations which will keep the speed of light c( and the forms of all laws) the same in all uniformly moving frames when the coordinates are used to calculate the ratio Dx/Dt or Dx' /Dt' which is the speed of light when x and t or x' and t' lie along the photon world line path at 45 deg to the x axis where x' and t' are skewed symmetrically relative to x and t?? Why does the scaling of the Galilean transformations result in the Lorentz transformations x'=Y(x-vt), t'=Y(t-(v/c^2)x} which keep the ratio Dx/Dt=Dx'/Dt'=c=the same in all such systems?
• I can see that scaling the Galilean transformations by Y means that bi0th x and t coordinates are multiplied by Y so we get
x' =Y(x-vt)=Y.x-Yv.t which is in analogy with the rotation of coordinate systemswhich has its coordinates transforming as x'=(cos Q).x+(sin Q).y where (cos Q) is the projection function of x onto the x
axis so that by analogy this would seem to correspond to Y which scales x onto the x'axis of the squashed system and where Yv seems to be analogous to sin Q which projects the y=ct coordinate onto the y'=ct' axis.,now I see how the circular functions can perform the orthogonal perpendicular projection BUT I do not see how Y or Yv can perform the parallel tilting(squashing) projections for the Lorentz transformations of the x coordinate onto the x'axis(Y is the projector function), or the ct corrdinate onto the ct' axis(projectorYv)
• How does the "stipulated condition" that the speed of light (as calculated from the coordinates ) be the same value of the ratios Dx/Dt and Dx/Dt' both be equal to c in all uniformly moving frames imply the particular way the coordinates transform between such relatively moving frames ,which transforrmations are called the Lorentz transformations and which are expressed in terms of the Lorentz factor
{1/sqrt(1-v^2/c^2)] which results from the invariance of the spacetime interval this invariance being equivalent to the condition of constant speed of light in all inertial frames.
The condition tha(t Dx/Dt=dx'/Dt' be a constant when calculated from the transformed coordinates of the frames tells how the coordinates have to be skewed(tilted) so that this condition Dx/Dt=Dx'/Dt'=c holds when the grid lines(coincidence and simultaneity lines) of the systems are tiled due to their relative uniform motion,
That is the invariance of dI^2=c^2dt^2-dx^2 shows how the coordinates have to be skewed(squashed) and tilted in order for this condition on the speed to hold .Then these transformations are gib=ven in terms of the Lorent factor Y.
x'=Y(x-B.ct)=Y(x-vt)=Y.x -Yv .t =Y.x--YB.ct
t'=Y(ct-B.x)=Y(ct-Bx)=Y.t-Y(v/c^2).x=Y.t-Y(v/c^2).x

B=v/c, Y=1/sqrt(1-v^2x/c^2)