Doubt regarding derivation of Lorentz Transformations.

  • I
  • Thread starter Kaguro
  • Start date
  • #1
221
57
I have just started learning the Special Theory of Relativity. While deriving, I am facing some problems. I obviously have made some kind of mistake while using the equations...
What is wrong if I don't use the time transformation equation in Event #2?
pic1.jpg



pic2.jpg
 

Attachments

  • pic1.jpg
    pic1.jpg
    41 KB · Views: 2,215
  • pic2.jpg
    pic2.jpg
    52.9 KB · Views: 695

Answers and Replies

  • #2
stevendaryl
Staff Emeritus
Science Advisor
Insights Author
8,942
2,931
Since y and z aren't involved, let's simplify to just talk about x and t.

Then a Lorentz transformation transforms a pair (x,t) into another pair, (x',t'). We're assuming that ##x' = Ax + Bt## and ##t' = Dx + Et##. Maybe it's clearer if we write them as functions:

##F_{x'}(x,t) = Ax + Bt##
##F_{t'}(x,t) = Dx + Et##

Then we have the assumptions:

Assumption 1: When ##x'=0##, ##x=vt##.

This means that the function ##F_{x'}(x,t)## satisfies ##F_{x'}(vt, t) = 0##. This implies that ##B = -vA##.

Assumption 2: When ##x=0##, ##x'=-vt'##.

This means that the functions ##F_{x'}(x,t)## and ##F_{t'}(x,t)## satisfy:

##F_{x'}(0, t) = -v F_{t'}(0, t)##.

So this implies that ##A\cdot 0 + B t = -v (D \cdot 0 + E t)##. So ##B = -vE##.

So you can conclude that ##E = A## and ##B = -vA##. But you can't conclude that ##D = 0##. To get ##D##, you need another assumption. That's usually the invariance of the speed of light:

Assumption 3: When ##x=ct##, then ##x' = c t'##.

In terms of functions,

##F_{x'}(ct, t) = c F_{t'}(ct, t)##
 
  • #3
221
57
Since y and z aren't involved, let's simplify to just talk about x and t.

Then a Lorentz transformation transforms a pair (x,t) into another pair, (x',t'). We're assuming that ##x' = Ax + Bt## and ##t' = Dx + Et##. Maybe it's clearer if we write them as functions:

##F_{x'}(x,t) = Ax + Bt##
##F_{t'}(x,t) = Dx + Et##

Then we have the assumptions:

Assumption 1: When ##x'=0##, ##x=vt##.

This means that the function ##F_{x'}(x,t)## satisfies ##F_{x'}(vt, t) = 0##. This implies that ##B = -vA##.

Assumption 2: When ##x=0##, ##x'=-vt'##.

This means that the functions ##F_{x'}(x,t)## and ##F_{t'}(x,t)## satisfy:

##F_{x'}(0, t) = -v F_{t'}(0, t)##.

So this implies that ##A\cdot 0 + B t = -v (D \cdot 0 + E t)##. So ##B = -vE##.

So you can conclude that ##E = A## and ##B = -vA##. But you can't conclude that ##D = 0##. To get ##D##, you need another assumption. That's usually the invariance of the speed of light:

Assumption 3: When ##x=ct##, then ##x' = c t'##.

In terms of functions,

##F_{x'}(ct, t) = c F_{t'}(ct, t)##


Wow! Thank you very much! You explained very nicely. I was finally able to derive the Lorentz Transformations.
 
  • #4
Orodruin
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
Gold Member
2021 Award
18,467
8,364
So you can conclude that ##E = A## and ##B = -vA##. But you can't conclude that ##D = 0##. To get ##D##, you need another assumption. That's usually the invariance of the speed of light:

Assumption 3: When ##x=ct##, then ##x' = c t'##.

In terms of functions,

##F_{x'}(ct, t) = c F_{t'}(ct, t)##

Just to highlight the difference to Newtonian mechanics. A different assumption (and an incompatible one at that) would be ##t' = t##, which directly would imply that ##D = 0## and ##E = 1## and therefore leads to the Galilei transformation
$$
x' = x - vt, \quad t' = t.
$$
It is important to note that the special principle of relativity is still valid in Newtonian mechanics, what really changes when going to SR is the assumption that the speed of light is invariant replacing the assumption that there is a universal time.
 

Related Threads on Doubt regarding derivation of Lorentz Transformations.

Replies
15
Views
988
  • Last Post
Replies
6
Views
2K
  • Last Post
Replies
12
Views
4K
  • Last Post
Replies
4
Views
1K
Replies
16
Views
5K
Replies
28
Views
2K
Replies
9
Views
1K
  • Last Post
Replies
8
Views
3K
  • Last Post
Replies
16
Views
15K
  • Last Post
Replies
3
Views
1K
Top