How Does Invariance Apply in Relativity Theory?

[timetraveller123]

Homework Statement

Homework Equations

The Attempt at a Solution

ok so for w' i am getting since the s' is only moving in x direction
##
\omega' = \omega \gamma (1 + \beta)
##
is this correct then i am having some trouble in dealing with the dot product to derive for
##
\bf {ck'}
##

 

[Orodruin]

It is not generally correct. Please show your work.

 

[timetraveller123]

It is not generally correct. Please show your work.

i am not getting what you mean
this is what i did
##
\omega = 2 \pi f\\
\omega' = 2 \pi f'
##
say an event occurs infront of moving observer then
##
f' = f \gamma (1+ \beta)
##

 

[Orodruin]

say an event occurs infront of moving observer

You cannot make this assumption.

 

[timetraveller123]

oh are you saying the event can occur above or below the x axis

 

[timetraveller123]

ok then letting the event be anywhere on the xy plane then
let v be the relative velocity between s' and s
and ##\theta## be angle formed in s and ##\theta'## the angle formed in s'
f be the frequency in s and f' be the frequency in s'

then
##
f = f_0\\
f' = f_0 \frac{\sqrt{1 - \frac{v^2}{c^2}}}{1 - \beta cos \theta'}

##
is this correct?

 

[Orodruin]

You are seriously overcomplicating things. All that is asked is how ##\omega'## and ##\vec k'## needs to be expressed in terms of ##\omega## and ##\vec k## for the expression ##\vec k \cdot \vec r - \omega t = \vec k' \cdot \vec r' - \omega' t'## to hold. You just need to make sure that this holds regardless of the values of ##t## and ##\vec x##.

 

[PeroK]

Homework Statement

View attachment 215123

Homework Equations

The Attempt at a Solution

ok so for w' i am getting since the s' is only moving in x direction
##
\omega' = \omega \gamma (1 + \beta)
##
is this correct then i am having some trouble in dealing with the dot product to derive for
##
\bf {ck'}
##

This looks like a maths problem to me. Does it really matter what these variables are? You're given a set of things (four-vectors etc.) with certain properties. I'm not sure you need to make any further assumptions about the things you are dealing with, other than the mathematical relationships you are given.

PS I see @Orodruin has just said much the same thing.

 

[timetraveller123]

You just need to make sure that this holds regardless of the values of ttt and ⃗xx→\vec x.

thats just what i am having problems with

This looks like a maths problem to me. Does it really matter what these variables are? You're given a set of things (four-vectors etc.) with certain properties. I'm not sure you need to make any further assumptions about the things you are dealing with, other than the mathematical relationships you are given.

the problem is i am not good at maths in relativity

please give me hints

 

[Orodruin]

Write out both sides of the equation, identify terms with the same coordinate factor ##t##, ##x##, ##y##, or ##z##.

 

[timetraveller123]

ok what does position vector mean like what is the r vector to event or to the moving obsever

 

[PeroK]

ok what does position vector mean like what is the r vector to event or to the moving obsever

Hint: this is a maths problem. You can forget "events" and "moving observers". Just focus on the equations now. The physics has been done by someone else: they are just asking you to do a bit of maths for them!

 

[timetraveller123]

Hint: this is a maths problem. You can forget "events" and "moving observers". Just focus on the equations now. The physics has been done by someone else: they are just asking you to do a bit of maths for them!

sir i understand what you are saying but i don't how to deal with the dot product how to separate the r' and k' vector when there is a dot between them thanks

 

[Orodruin]

sir i understand what you are saying but i don't how to deal with the dot product how to separate the r' and k' vector when there is a dot between them thanks

I am sorry, but if you have problems with that part you are not mathematically prepared to tackle relativity. Before you can deal with relativity, you need to become more proficient in vector algebra and vector analysis.

 

[timetraveller123]

I am sorry, but if you have problems with that part you are not mathematically prepared to tackle relativity. Before you can deal with relativity, you need to become more proficient in vector algebra and vector analysis.

what level vector algebra is needed because i know the basics only

 

[timetraveller123]

@Orodruin
sir please tell what do i need to learn because i am more than happy to learn please guide me through this problem and this problem