Gamma factor in legth and mass demostration

[Littlepig]

Homework Statement

I'm studing relativity(on extraschool) and known about gamma factor influences time, after some tries, I demostrate that if A is at Va, in relation to B: "light distance"=sqrt("light distance for B"^2-"distance traveled by A"^2)

where we say that "light distance"=c*"ta"; "light distance for B"=c*"tb"; and "distance traveled by A"=Vatb resolving and we demonstrate it...i gess you know what i mean...

now my problem is who to demonstrate that leght and mass are too influenced by gamma factor.

Homework Equations

wish i know then...

The Attempt at a Solution

don't have a tought...:X

 

[Hootenanny]

You mean you wish to prove length contraction?

 

[Littlepig]

You mean you wish to prove length contraction?

indeed, and mass, but by maths method, not by "concept" method, what is the begining? what is the permiss that is missing me to acept that Y can too interfer in mass and leght(by math method)

 

[Hootenanny]

Length contraction follows directly from time dilation. Time dilation can be 'proved' using a simple thought experiment. Length contraction follows directly from the lerentz transformations, so there is little in the way of a 'mathematical' proof, as far as I know.

 

[Littlepig]

Length contraction follows directly from time dilation. Time dilation can be 'proved' using a simple thought experiment. Length contraction follows directly from the lerentz transformations, so there is little in the way of a 'mathematical' proof, as far as I know.

yes yes, till there i agree with it, but i wnat to demonstrate it mathematicaly...

ok, my point is:

for have time dilatation we have that:

2 obeservers, A and B, A is moving at speed Va and B is still.
"A" have a light clock(2 perfect mirrors, 1 photosensor, 1 laser making a 90º to mirrors).

as light travels at same speed at diferent referencials, we have that B will see the light moving not at 90º but at a diferent angle.
A will jut see light moving up and down like nothing was hapening...

now, from trignometry, $$c_1=sqrt(H^2-c_2^2)$$

as $$V=d/t$$,

we have that:
distance that light takes, measured by B is H,
distance travel by light, measured by A is $$c_1$$;
distance travel A, measured by B, is $$c_2$$;

so as $$d=V \cdot t$$;

$$t_A \cdot c= \sqrt{(t_B \cdot c)^2-(t_b \cdot V_a)^2}<=>t_B= \frac{t_A}{ \sqrt{1 - \frac{v^2}{c^2}}}$$

now, i demonstrated with maths the time dilatation...how i demonstrate leght contraction and mass dilatation?

 

[Littlepig]

how i demonstrate leght contraction and mass dilatation?

still looking for some clue, I've search in wiki, but everything is already demonstrated, they only put the formulas there...the rest is explication of then...:(

 

[robphy]

consider a light clock [whose separation of mirrors is] oriented in the direction of motion, rather than perpendicular...
..and don't forget to use the principle of relativity.

 

[Littlepig]

yes yes, till there i agree with it, but i wnat to demonstrate it mathematicaly...

ok, my point is:

for have time dilatation we have that:

2 obeservers, A and B, A is moving at speed Va and B is still.
"A" have a light clock(2 perfect mirrors, 1 photosensor, 1 laser making a 90º to mirrors).

as light travels at same speed at diferent referencials, we have that B will see the light moving not at 90º but at a diferent angle.
A will jut see light moving up and down like nothing was hapening...

now, from trignometry, $$c_1=sqrt(H^2-c_2^2)$$

as $$V=d/t$$,

we have that:
distance that light takes, measured by B is H,
distance travel by light, measured by A is $$c_1$$;
distance travel A, measured by B, is $$c_2$$;

so as $$d=V \cdot t$$;

$$t_A \cdot c= \sqrt{(t_B \cdot c)^2-(t_b \cdot V_a)^2}<=>t_B= \frac{t_A}{ \sqrt{1 - \frac{v^2}{c^2}}}$$

now, i demonstrated with maths the time dilatation...how i demonstrate leght contraction and mass dilatation?

ok, after some tries, i think i reach a point it's correct, however, i would like a confirmation of what I've done.

so, to demonstrate a leght contration, I used this logic:

$$t_B= \frac{t_A}{ \sqrt{1 - \frac{v_a^2}{c^2}}}, V=\frac{d}{t}<=>t=\frac{d}{v}$$
with d as the leght travel by "a".

so, as v_a is equal to both observers,

$$\frac{d_b}{v_a}=\frac{d_a/v_a}{ \sqrt{1 - \frac{v_a^2}{c^2}}}$$

what is equal to:

$$\frac{d_b}{v_a}=\frac{d_a}{v_a \cdot \sqrt{1 - \frac{v_a^2}{c^2}}}$$

what is equal to:

$$d_b=\frac{d_a}{\sqrt{1 - \frac{v_a^2}{c^2}}}$$

what is equal to:

$$d_a=d_b \cdot \sqrt{1 - \frac{v_a^2}{c^2}}}$$

so, to observer "a", who is moving, leght is everytime lower than to "b", and convertor factor is "Y"

am I right??


to demonstrate mass dilatation, I dout it's correct by my though, however, i putted it here :

$$d_a=d_b \cdot \sqrt{1 - \frac{v_a^2}{c^2}}}$$
and,
$$D= \frac{m}{V}$$
where m is mass, D is density, and V is Volume

so, considering a rectangular object,

$$m= D \cdot (H \cdot W \cdot d)$$
where H is High, w is Width, and d is Leght

considering m_a the mass of "a" for the observer "a" and m_b the mass of "a" for the observer "b"

for moving observer "a"
$$m_a= D \cdot H \cdot W \cdot d_a$$

for rest observer "b":
$$m_b= D \cdot H \cdot W \cdot \frac{d_a}{\sqrt{1 - \frac{v_a^2}{c^2}}}}$$

so, taking D off:

$$\frac{m_a}{H \cdot W \cdot d_a}=\frac{m_b}{H \cdot W \cdot d_a} \cdot \sqrt{1 - \frac{v_a^2}{c^2}}}$$

as H, W, D and d_a are equal to both observers,

$$m_a=m_b \cdot \sqrt{1 - \frac{v_a^2}{c^2}}}$$

so,

$$m_b=\frac{m_a}{\sqrt{1 - \frac{v_a^2}{c^2}}}}$$

can it be done with this logic? or we need to use another way??

thanks in advance, and hope wasn't too annoying