Unraveling the Gamma Factor Equation

[Infamo]

I'm new to the whole relativity thing, and I've been reading the book
E=mc^2, The Equation That Changed the World. I don't know if any of you have read the book, but, in the book, give the gamma factor equation, gammefactor=1/sqrt[1-(v/c)^2], now, here are my questions.
1.) What really does the gamma factor represent?
2.) How did einstein come up with this equation?
3.) In the book, (physicist named Haller, Einstein, and Newton are talking) they go to find out how fast you would have to go to reach the center of the galaxy(Milkyway), in 30years(with time dialation), he says that the gamma factor would have to be 30,000/30, which is 1,000(corresponding gamma factor). How he doesn't state where he got 30,000 from, but I can see where 30 came from(30 years) but not 30,000. Now it doesn't state how far the person is traveling(earth to the center of the galaxy), that is all the information that is given.
Any help would be appreciated,
Thanks

 

[da_willem]

Hi Infamo, welcome to PF!

I read the book, and found it a pretty entertainig read. About your questions:

1) It tells you how time and length change as a function of velocity. When you see something going by with a velocity v you wil measure its length (in the direction of motion!) contracted as compared to the objects 'rest length'. This 'Lorentz contraction' can be written in a formula:

$$L=L_0/\gamma$$ with $$\gamma =\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$

With $L_0$ the objects rest length.

Also the time you assign to a certain time interval (t) will differ from what someone else will asign to it ($t_0$) who is moving with a velocity v as compared to you. Your time will 'slow dow'. This 'time dilation' can be written in a formula:

$$t=\gamma t_0$$

2) I don't think Einstein was the first to derive these formulas. For example the length contraction will probably not be named after Lorentz for nothing. But the easiest way to derive them is probably by means of a 'light clock'. The main assumption you need is that the speed of light is the same for everyone who measures it (and is not accelerating or in a gravitational field). See for example: http://library.thinkquest.org/C008537/relativity/math/math.html

3) For the third question you have to realize that when you are moving in your spaceship not only will someone on Earth wil measure you spaceship contract, but also you will see the Earth and the entire universe (in the direction of your motion) contract! So the distance to the center of the universe will no longer be that far away, but you have to go very fast wil you be able to reach it in 30 years, you have move with a velocity very close to that of the speed of light. The amount of time it takes to get there is the (contracted) distance ([itex]L_0/\gamma[/tex]) divided by your velocity:

[tex] 30 years = \frac{L_0}{\gamma v}[/itex]

Now the distance to the galactic center is ~2,5E20 m, solving for gamma with v~c yields ~10^3 (convert 30 years to seconds!)

 

[da_willem]

Note: $$\gamma \approx \frac{L_0 / c}{30 years}$$

Now $L_0 / c$ is the time it takes light to reach the center of the galaxy. So the 30.000 you read in that book probably comes from the distance to the center of the galaxy in 'lightyears'.

 

[Infamo]

Now that I do understand the gamma factor equation, I don't really understand space-time contraction. Can you point me to some good sources or you yourself help me out. Thanks

 

[da_willem]

Did you read the link I gave you. It is a good introduction if you're also interested in the mathematics behind special relativity. And for more information just search with Google with search terms like Lorentz contraction. You're the one that knows what sites you enjoy reading...