How does SR work at speeds greater than c?

[aychamo]

Does SR work/holdup if something were to travel at multiples, instead of fractions, of c?

For example, if you travel at .865c for 1 hour, you age one hour, while everyone else on Earth ages 2 hours. But if you travel at 10c for 1 hour, you age one hour but everyone on Earth ages .. I don't know the math :) But I'm sure it's somewhere around 10 hours?

Obviously this is science fiction, but it is fun to talk about nonetheless.

Continuing, in Star Trek we learned that people in the future wear spandex uniforms and that they can travel at warp speeds, which are, I presume, great multiples of C.

So if you were to travel a great distance at a warp speed, but in a very short amount of time, what would the effects be? On initial thought, it seems that if you travel for any significant amount of time at a huge multiple of c then anywhere you go will be so greatly aged by the time you get there that your trip there would be pointless.

When I think of this in my head, I see it as the guy who took Einstein's equations and used them to think up black holes, although that was an original thought and I'm sure what I'm writing about has been covered before somewhere else.

So does calculating the gamma factor, with the 1/sqrt(..whatever) equation hold up when using multiples of c, instead of fractions? If I wanted to put in some figures to test my ideas with different values in the above paragraph, what simple equations could I use?

Thank you
Aychamo

 

[UltanByrne]

The easiest equation (That I know of) to see the relativistic effects of high velocities are the Lorentzian transformation equations, specific to your question the equation for Time Dilation:

where v = your velocity,
c = speed of light.

(Picture from Wolfram's Science world)

So you want to know what happens when v > c eh? Let's c:
The First thing we do to simplify is make up a new scale, we are going to say that c = 1. (so we don't have to deal with really big numbers, the actual numbers we get don't really matter, just relations)

Lets also say that v = 2c, in this example you will be traveling at twice the speed of light.

Lets plugin:

= 1 / Sqrt(1 - v^2) -> The modified equation with c = 1
= 1 / Sqrt(1 - 4) -> 2 squared = 4
= 1 / Sqrt(-3) -> Ding Ding Ding, your mathy sense should be tingling, you can't get a regular answer when you have a negative number under a root sign (try it on a calculator), to deal with this, mathematicians invented the concept of imaginary numbers (i = Sqrt(-1)), so given that this equation is for time dilation, what you are dealing with here is some sort of freaky deaky imaginary time (Star trek would be proud),

So in conclusion, when you plug any number greater than c (1 in our system of units) into the Lorentzian equation of time dilation you get an imaginary answer, less than c and you get good ol' relativistic modifications, and at c itself you get Sqrt(0)!

Although this is all quite exciting, you really shouldn't start comparing yourself to Schwarzschild (Found the first solution set to Einstein's field equations -> Black Hole), but you're well on your way to follow in his path :)

Hope I've been helpful!

Ultan

 

[aychamo]

Hey man!

Good answer! That kinda sucks there is no way to play around with time dilation with regards to speeds greater than c!

Thank you though,
Aychamo

 

[DW]

Originally posted by aychamo
Hey man!

Good answer! That kinda sucks there is no way to play around with time dilation with regards to speeds greater than c!

Thank you though,
Aychamo

There is, just not within the context of "special" relativity. Consider the following modification of Alcubierre's spacetime where a ship has the hypothetical ability to modify the surrounding spacetime geometry. Let's say that the interval for the ship frame will be
$$ds^2 = (A^2 - \beta ^{2}g^{2})dct^2 - 2\beta gdctdz - dz^2 -dr^2 - r^{2}d\phi ^2$$
were A will be 1 in the ship and a constant far from it, and g will be zero in the ship and 1 far from it. $$\beta$$ respresents v/c that the ship crew observe remote stars to be passing by and A is a time lapse function similar to the special relativistic $$\gamma$$. What you choose the far value of A to be determines the time dilation and a combination of the bahaviors of A and g determine the matter requirements to form the modified spacetime geometry.

 

[UltanByrne]

aychamo, glad I could help!

DW, interesting... I have to admit I've never heard of Alcubierre's spacetime!

Ultan

 

[DW]

Originally posted by UltanByrne
aychamo, glad I could help!

DW, interesting... I have to admit I've never heard of Alcubierre's spacetime!

Ultan

In all seriousness its called the Alcubierre warp drive spacetime. The expression I wrote is a sleight modification of his. The difference being that he wrote it for a remote observer's coordinates and inserted a lapse function into the expression for with those coordinates whereas the expression I wrote is the expression for the ship frame coodinates with the lapse function inserted into the expression for with those coordinates. With a lapse function not globally 1, the one expression isn't a mere frame transformation of the other.