Exploring the Twin Paradox: A 1,000 Word Essay

[chroot]

Why you are dividing $$\gamma$$ by $$.99 * c$$.
Also, when I try to convert 3 light years to actual numbers and then change the .99 * c to 184140 the equation doesn't work. So I don't know what 3 light years or even c represents if it doesn't work.

Basically what I am trying to say is, if I were to do this on an actual calculator, not google.com how would I write it (with numbers included). I can't put a c so I assume its 186000, I can't write 3 light years so I assume its 558000, but putting these in they don't calculate to the 156 days.

I hope you understand what I am getting at.

Okay, you should never mention miles again. Ever. If you ever even think of miles again, you're doing something wrong. Do not call c 186282 miles/second. Do not call c 300,000 km/sec either. Call c one light-year per year.

That's right -- c is just one in units where you use light-years and years as your units of length and time.

Now, gamma's easy to calculate, as I showed you. When you're considering velocities as fractions of c, then the c's cancel out. When you're considering 0.99c, for example, the term (0.99c)^2/c^2 can be simplified to just 0.99^2. Gamma is then just 1/(1-0.99^2), which you can do on any calculator. No miles. No unit conversions. If you ever attempt to use miles ever again, you're doing something wrong.

Now, gamma is dimensionless. It does not have units of time, or length, or anything. Gamma is just a number, a multiplicative factor.

When you want to use gamma in the time-dilation or length-contraction equations, leave the times in years, and the lengths in light-years.

If the twin is traveling three light-years at 0.99c, that three light-years will be contracted to 3/gamma light-years (about 0.423 light-years). Simple division. No miles.

The twin will cross those 3/gamma light-years in (3/gamma)/0.99c. Remember in our units c is one-light yer per year. Here are the units:

$$\frac{3 \,\textrm{light-years}}{\gamma} \, \cdot \, \frac{1}{0.99 \,\textrm{light-years per year}} = 0.427 \,\textrm{years}$$

Or, about 156 days.

- Warren

 

[Prague]

Ah, I understand this now. Thank you, hopefully this will get me by on my paper. I'll use this to prove there is a time difference, then I can say that the paradox is which twin is aging slowly because each twins reference is that the other twin is slowing down. And the simple answer is, the traveling twin ages slower because he leaves the the constant inertial rate of X, correct?

 

[chroot]

You can use the time-dilation equations to show what the paradox means. The time-dilation equation is symmetrical, meaning that both the earth-bound and the spaceship-bound twin sees the others' clock as running slowly. You can show some numbers to make the concept of time dilation more concrete.

The resolution of the paradox comes from the realization that, for the twins to re-unite, one of them must turn around. The twin that turns around, of course, is the one on the spaceship. He has to push buttons on his console to fire his engines. He feels the forces of the engines. His coffee cup gets knocked over, etc. After he's turned around, he's no longer in the same frame of reference that he was when he was moving away from the Earth. The introduction of this new third frame breaks the symmetry of the paradox. When you solve the problem will all three frames considered, the paradox is no longer a paradox: All observers will see the spaceship-twin as aging less than the Earth-twin.

- Warren

 

[Prague]

Alright, thanks a lot I understand everything now. Now on to writing.

Thank to everyone for the help.

 

[chroot]

Good luck with the paper!

- Warren