# Coordinate Transformation in Special Relativity with Linear Algebra Part A

## Homework Statement

In the figure, let S be an inertial frame and let S'
be another frame that is
boosted with speed v along its x'-axis w.r.t. S, as shown. The frames are pictured
at time t = t0 = 0:
A) Find the Non-relativistic transformation (Galilean Transformation) between the two frames

## Homework Equations

- Basic transformation in Linear Algebra:
Ax = x'
v = dx/dt

## The Attempt at a Solution

I did forget the v = dx/dt part in A, but i think it is merely a change of notation that is all,
for example, instead of writing v i will write de/dt.

The First attachment is the Figure you need to look at, the second attachment is my attempted solution as a pdf, i really want to learn Physics Forum, and i don't want the "answers" i want to see if i am correct or not, if not, just point where i went wrong and give a little hint.
Thank you

#### Attachments

• Figure1.jpg
9.8 KB · Views: 349
• Attempted Solution.pdf
175.1 KB · Views: 159

## Answers and Replies

The basic idea is correct, but it seems there is some confusion of velocity and position in the solution. I suggest that you introduce frame S'', which is rotated with regard to S but is stationary. Then find transformations between S and S'' and S' and S'', and combine them.

I already done a stationary rotation. I will try it again and post.

Ok dear forum users and @voko:
i have done in in three parts:
- Origins overlap and just a mere rotation
- fixed distance and rotation
- variable distance and rotation

can you check all three? probably takes around 2-5 mins. No final answers, just tell me where i went wrong. if not, say that i am correct (give me a cookie!)

umm, this is all Part A of a qeustion that well, when you see part A - Variable Distance i need to transform it relativisticlly.
I have no idea how to approach that (i used galilean transformation on all 3, can you give me a "solid" hint on how to approach with with lorenz transformation?)

Check if my answers are correct, if they are, i will post the next question, solve it and come back here to check answers.

thank you physicsforums

ps: i've redone what you said i should do voko, i did them all

#### Attachments

• Rotation with origins = 0.pdf
106.6 KB · Views: 128
• Rot with Fixed Dist.pdf
173.8 KB · Views: 158
• Rot With variable Distance.pdf
286.9 KB · Views: 141
There are two kinds of rotation: rotation of an object in a fixed frame, and rotation of the frame with a fixed object. They have similar rotation matrices; in fact, one is the inverse of the other. Your matrix is that of object rotation, while you need one for frame rotation.

Here is how you can deduce the correct matrix easily: let's say the object is at (X, Y). X = R cos α, Y = R sin α, where R is the distance to the object and α is its polar angle. In a frame rotated by angle θ with respect to xy, the distance to the object does not change, but the polar angle is (α - θ), so X' = R cos (α - θ), Y' = R sin (α - θ). Using trigonometric identities, you should get X' = X cos θ + Y sin θ, Y' = X sin θ - Y cos θ.

I am not sure why you are considering rotation with fixed translation, it does not really help you here.

Transformation to a rotated frame moving with constant velocity v along its X' axis is best done by considering the stationary frame S'', whose origin O' = O, and which is rotated by the same angle. Then (X'', Y'') = rot(X, Y), where rot is the matrix discussed above. (X', Y', T') = (LorentzX(X'', t, v), Y'', LorentzT(X'', t, v)).

thank you for the help, i will look into the "frame rot" vs "object rot", and i think i understood the matrix:
_ cos(a) sin(a)
|
|_sin(a) -cos(a)

? rite? i am just guessing, as i need to sleep now :)
i did the fixed distance so i can change only one parameter which is that it is exactly the same equations but the distance isn't fixed... but now varies with time.

i thought my approach was good,... wasn't it? i will repost fixed matrices tomorrow

Ive done the deduction and i got this instead:

x' = xcosθ + ysinθ
y' = ycosθ - xsinθ

aka, y' was -1 your y'.

I am currently almost done with their respective lorentz transformation... post it shortly afterwards

Yes, I made a mistake, you result is correct so far. See, everybody gets confused about these rotations :)

since ur already here... should i get the lorentz transformation of V, too?
becuse without it it seems easier :P ... but i really think that i need the transformation of v, since v acts on both X and Y.. (in S)

it's only in X' in S'

If v is velocity, you don't need to transform it. You know in advance it will be aligned with OX' - that's why you rotate the frame to begin with.

then i think i am done, give me 5 mins to copy the stuff to my "special" :P paper and attach as PDF, i will include some other inference i had on V, please tell me if they are true or false.. i will be more specific in the pdf

I think i've done this question, here is my latest attempt,
i also have some "True/False" questions, that are there i made them up... because i got confused, and i need some clarification. they are labeled i, ii, iii , iv (i think i forgot iv but it's the last thing i wrote :P )

#### Attachments

• A Boosted frame with rotation.pdf
258.7 KB · Views: 132
You have two transformations, which look correct. But you have only two frames denoted in your solution. You can't have two transformations between two frames, it does not make sense. This is why you are confused. You are almost there, but you need to label all the THREE involved frames and the corresponding quantities correctly.

orrite, i will try again with S being our normal rf and S' the fixed rotated and S'' the moving rotated