# Lorentz transformations

toph
Question
Show that, with V = 4/5c, the Lorentz transformation of the equations, t^prime = y(V) (t-(v/c^2)x) and x^prime = y(V) (x-Vt). (where y(V) = the Lorentz factor).

can be written as

ct^prime = 5/3ct - 4/3x

and

x^prime = 5/3x - 4/3ct

Relevant equations
y(V) = 1/(sqrt1-(V/c)^2)

The attempt at a solution
I have calculated y(V) = 5/3 (if V = 4/5c)
and i can see how the left hand term in each equation becomes 5/3ct and 5/3x respectivley. But i cant figure where the 4/3 term comes from?? or how to derive it?

## Answers and Replies

neutrino
The second term on the RHS in both equations contain a 'v', right?

Staff Emeritus
Gold Member
$$\left( \frac{5}{3} \right) \left( \frac{4}{5} \right) = \frac{4}{3}.$$

toph
Ahh... the penny drops. thank you

toph
A further Lorentz transformation problem.

The question i have is.

Use these Lorentz transformations ct'=5/3ct-4/3x and x'=5/3x-4/3ct. to determine the (ct', x') coordinates, in meters, that observer O' assigns to events e1 and e2.

Relevent equations and information.

from a previous question i have determined the coordinates of the events in the rest frame of observer O to be e1 = (0, 240)m, e2 = (60, 240)m.

My attempt at answer

Using the given Lorentz transformations i have found for event e1 as observed by O' is.

e1 = (ct', x') = (5/3ct-4/3x, 5/3x-4/3ct) = (-180, 144)m
e2 = (ct', x') = (5/3ct-4/3x, 5/3x-4/3ct) = (-144, 99) m

However i feel uncomfortable with these answers, but cannot put my finger on why? Please can some one check my results?

thank you

toph
Further to my last post, the reason i feel uncomfortable is that surely the x component of the coordinates should be the same for both events? I can check this via the Lorentz length contraction formula, which gives l = lo/y(V) =144?

neutrino
It seems to be errors in arithmetic to me.

Further to my last post, the reason i feel uncomfortable is that surely the x component of the coordinates should be the same for both events? I can check this via the Lorentz length contraction formula, which gives l = lo/y(V) =144?

Surely, then, both observers must be referring to the same reference frame.

toph
It seems to be errors in arithmetic to me.

Surely, then, both observers must be referring to the same reference frame.

yep your right i have just spotted my error. i think i have also calculated the transformed event coordinations incorrectly as well?

Last edited:
neutrino
Surely, then, both observers must be referring to the same reference frame.
Sorry about my previous post. I misread your statement, and gave a stupid reply (eyes can play tricks on you late at night!). I really cannot say whether the x coordinates of the two events in the O frame are the same or not, unless I have more information. But if you have managed to find you errors in spite of my comment, then well and good.