Checking the Lorentz transformation

[QipshaqUli]

Homework Statement

This seemed at first glance very easy. But there appeared some confusion.

A is moving to the right with velocity v with respect to B. The proper time for A is $t_a=t_b\sqrt{1-v^2/c^2}$. And B is moving to the right with velocity u with respect to C. Proper time for B $t_b=t_c\sqrt{1-u^2/c^2}$. $t_a$ can be found by $t_c$ then: $t_a=t_c\sqrt{1-u^2/c^2}\sqrt{1-v^2/c^2}$ then. Futher, by using the law of addition of relativistic velocities one can find the relative velocity of A with respect to C: $w=\frac{u+v}{1+\frac{uv}{c^2}}$. And defining proper time for A by $w$ I found $t_a=t_c\frac{\sqrt{1-u^2/c^2}\sqrt{1-v^2/c^2}}{1+\frac{uv}{c^2}}$ which is different from previous one.

Relevant Equations

$$t_a=t_c\frac{\sqrt{1-u^2/c^2}\sqrt{1-v^2/c^2}}{1+\frac{uv}{c^2}}$$

Homework Statement: This seemed at first glance very easy. But there appeared some confusion.

A is moving to the right with velocity v with respect to B. The proper time for A is $t_a=t_b\sqrt{1-v^2/c^2}$. And B is moving to the right with velocity u with respect to C. Proper time for B $t_b=t_c\sqrt{1-u^2/c^2}$. $t_a$ can be found by $t_c$ then: $t_a=t_c\sqrt{1-u^2/c^2}\sqrt{1-v^2/c^2}$ then. Futher, by using the law of addition of relativistic velocities one can find the relative velocity of A with respect to C: $w=\frac{u+v}{1+\frac{uv}{c^2}}$. And defining proper time for A by $w$ I found $t_a=t_c\frac{\sqrt{1-u^2/c^2}\sqrt{1-v^2/c^2}}{1+\frac{uv}{c^2}}$ which is different from previous one.
Homework Equations: $$t_a=t_c\frac{\sqrt{1-u^2/c^2}\sqrt{1-v^2/c^2}}{1+\frac{uv}{c^2}}$$

$t_c$ then: $t_a=t_c\sqrt{1-u^2/c^2}\sqrt{1-v^2/c^2}$

[Note from mentor: please see https://www.physicsforums.com/help/latexhelp/ for the proper delimiters for LaTeX code on this forum. I have edited your post accordingly, in order to make your equations properly visible.]

 

[Orodruin]

You have a problem with the relativity of simultaneity. When you use the time dilation formula for B relative to A you are using the simultaneity of A. Similarly, when you use the time dilation formula for C relative to B you are using the simultaneity in the rest frame of B. Thus, the combination will not correspond to the time dilation of C relative to A, which exclusively is based on simultaneity as defined by A.

 

[PeroK]

Homework Statement: This seemed at first glance very easy. But there appeared some confusion.

A is moving to the right with velocity v with respect to B. The proper time for A is $t_a=t_b\sqrt{1-v^2/c^2}$. And B is moving to the right with velocity u with respect to C. Proper time for B $t_b=t_c\sqrt{1-u^2/c^2}$. $t_a$ can be found by $t_c$ then: $t_a=t_c\sqrt{1-u^2/c^2}\sqrt{1-v^2/c^2}$ then. Futher, by using the law of addition of relativistic velocities one can find the relative velocity of A with respect to C: $w=\frac{u+v}{1+\frac{uv}{c^2}}$. And defining proper time for A by $w$ I found $t_a=t_c\frac{\sqrt{1-u^2/c^2}\sqrt{1-v^2/c^2}}{1+\frac{uv}{c^2}}$ which is different from previous one.
Homework Equations: $$t_a=t_c\frac{\sqrt{1-u^2/c^2}\sqrt{1-v^2/c^2}}{1+\frac{uv}{c^2}}$$

Homework Statement: This seemed at first glance very easy. But there appeared some confusion.

A is moving to the right with velocity v with respect to B. The proper time for A is $t_a=t_b\sqrt{1-v^2/c^2}$. And B is moving to the right with velocity u with respect to C. Proper time for B $t_b=t_c\sqrt{1-u^2/c^2}$. $t_a$ can be found by $t_c$ then: $t_a=t_c\sqrt{1-u^2/c^2}\sqrt{1-v^2/c^2}$ then. Futher, by using the law of addition of relativistic velocities one can find the relative velocity of A with respect to C: $w=\frac{u+v}{1+\frac{uv}{c^2}}$. And defining proper time for A by $w$ I found $t_a=t_c\frac{\sqrt{1-u^2/c^2}\sqrt{1-v^2/c^2}}{1+\frac{uv}{c^2}}$ which is different from previous one.
Homework Equations: $$t_a=t_c\frac{\sqrt{1-u^2/c^2}\sqrt{1-v^2/c^2}}{1+\frac{uv}{c^2}}$$

$t_c$ then: $t_a=t_c\sqrt{1-u^2/c^2}\sqrt{1-v^2/c^2}$

[Note from mentor: please see https://www.physicsforums.com/help/latexhelp/ for the proper delimiters for LaTeX code on this forum. I have edited your post accordingly, in order to make your equations properly visible.]

Instead, why not have $B$ moving to the left relative to $C$? The same formulas apply, but now frame $A$ and $C$ are the same frame, yet you have their clocks time dilated relative to each other.