Can the Lorentz Factor Approach 0i as v Exceeds c for FTL Reference Frames?

[Chris Miller]

As v approaches c, the Lorentz factor approaches infinity. The math and physics is well understood and observed. Is it true that, just mathematically, as v exceeds c the Lorentz factor approaches 0i for imaginary time constriction?

 

[PeterDonis]

imaginary time constriction

What is "imaginary time constriction"?

Is it true that, just mathematically, as v exceeds c the Lorentz factor approaches 0i

Have you tried to work this out mathematically? What does the math tell you?

 

[Nugatory]

As v approaches c, the Lorentz factor approaches infinity. The math and physics is well understood and observed. Is it true that, just mathematically, as v exceeds c the Lorentz factor approaches 0i for imaginary time constriction?

The Lorentz transforms are derived starting from assumptions that are equivalent to assuming that $v$ must be less than $c$. Thus, the complex numbers that appear when you try plugging in a speed greater than $c$ tell you nothing except that you're using the formula in a situation where it doesn't apply.

 

[Chris Miller]

Thanks for your answers. By "imaginary" I just meant the square root of a negative. By "constriction" I meant the opposite of dilation. I assumed that if time dilates as the Lorentz factor approaches infinity, then it must constrict as it approaches 0 (times i). The math is simple: 1/(n*i) as n approaches infinity. When you assume...

 

[Ibix]

Thanks for your answers. By "imaginary" I just meant the square root of a negative. By "constriction" I meant the opposite of dilation. I assumed that if time dilates as the Lorentz factor approaches infinity, then it must constrict as it approaches 0 (times i). The math is simple: 1/(n*i) as n approaches infinity. When you assume...

I think @Nugatory's answer is best. It doesn't make sense to do this. When you insert $v>c$ you may be defining some kind of coordinate system, but it doesn't have the naive interpretation you are trying to put on it.

 

[PeterDonis]

By "constriction" I meant the opposite of dilation. I assumed that if time dilates as the Lorentz factor approaches infinity, then it must constrict as it approaches 0 (times i).

A multiple of i is not the "opposite" of a real number in any useful sense, so I don't think this is a valid line of reasoning.

 

[Chris Miller]

Are you saying complex numbers have no use in physics? I wrote "imaginary time constriction." Not being understood, or applicable, might not make it meaningless. I'll admit to having a very weak grasp of complex numbers. Like even a simple statement like 2i < 3i I'm not sure is true, or if the < operator's use in it is valid.

 

[PAllen]

Which is greater, north or east? That’s silimar to trying compare real and imaginary numbers.

 

[PeterDonis]

Are you saying complex numbers have no use in physics?

No. I'm saying that you can't just wave your hands and say a complex number for the $\gamma$ factor means what you are saying it means.

I'll admit to having a very weak grasp of complex numbers. Like even a simple statement like 2i < 3i I'm not sure is true, or if the < operator's use in it is valid.

Your hesitation is warranted. Unlike real numbers, complex numbers are not totally ordered, so the comparison operators don't make sense for them the way they do for reals.

 

[waves and change]

Are you saying complex numbers have no use in physics? I wrote "imaginary time constriction." Not being understood, or applicable, might not make it meaningless. I'll admit to having a very weak grasp of complex numbers. Like even a simple statement like 2i < 3i I'm not sure is true, or if the < operator's use in it is valid.

A prerequisite so to speak for the Lorenz transform is that v<c... therefore that restriction must be assumed if you want to even consider the factor itself

 

[Chris Miller]

Thanks all. I'm going to google complex numbers. I understand they can't be compared to reals, but would've thought they lay on a number line in some order. I understand the Lorentz transform's intended usage and restrictions re SR, but as a purely mathematical equation, its only limitation to me would seem to be that v <> c.

Ha ha Peter. Didn't mean to wave my hands. Maybe it's my Italian roots. Also I'm not sure what I'm saying by "imaginary time contraction." What do you imagine I'm saying?

 

[PeterDonis]

I understand they can't be compared to reals, but would've thought they lay on a number line in some order.

They don't. They lie in a plane. Points in a plane do not have a linear order.

I'm not sure what I'm saying by "imaginary time contraction." What do you imagine I'm saying?

I don't think you're saying anything meaningful at all.

 

[stevendaryl]

The Lorentz transformations are used to describe the relationship between coordinates used in different inertial frames. Special Relativity doesn't assume the existence (or meaningfulness) of two inertial frames whose relative speed is faster than the speed of light. So the imaginary values of $\gamma$ don't apply to any situation covered by SR.

There might be a way to extend SR to describe FTL reference frames, but SR certainly doesn't have them.