Non-Riemannian geometry for inverse metric form

[Eugene Shubert]

Is it possible to invent a non-Riemannian geometry to justify the existence of a "metric" of the form:

1/ds^2 = 1/dt^2 – 1/(dx^2 + dy^2 + dz^2)

Eugene Shubert
http://www.everythingimportant.org/relativity

 

[schwarzchildradius]

isn't that just
ds² = dt²- (dx²+dy²+dz²)
?

 

[Eugene Shubert]

No, of course not. Remember your elementary algebra.

Eugene Shubert
http://www.everythingimportant.org/relativity

 

[Brad_Ad23]

Umm, I would agree with schwarzchildradius here. Just multiply through and suddenly you get rid of the nasty fractions.

 

[schwarzchildradius]

Yes I do remember elementary algebra, good for me. you can invert that equation.

 

[plus]

Would you like to expain in more detail how you think you can invert that fraction to get the required result?

 

[Considering]

I thought flipping a fraction such as 1/3^2 would result in 3^-2.

Doesn't it change the exponent?

 

[suffian]

C'mon guys...

1/ds² = 1/dt² – 1/(dx² + dy² + dz²)
1/ds² = (dx² + dy² + dz² - dt²)/[ (dx² + dy² + dz²)(dt²) ]

ds² = [ (dx² + dy² + dz²)(dt²) ]/(dx² + dy² + dz² - dt²)

Which just doesn't look any cleaner.

edit: changed to using integrated superscript.

 

[plus]

Originally posted by suffian
C'mon guys...

1/ds² = 1/dt² – 1/(dx² + dy² + dz²)
1/ds² = (dx² + dy² + dz² - dt²)/[ (dx² + dy² + dz²)(dt²) ]

ds² = [ (dx² + dy² + dz²)(dt²) ]/(dx² + dy² + dz² - dt²)

Which just doesn't look any cleaner.

edit: changed to using integrated superscript.

That was my point.

 

[Eugene Shubert]

Let me suggest the physical meaning to the expression above.

I’m thinking of ds as an invariant that represents a differential increment of proper time. That would imply that the total amount of elapsed proper time t' would equal t/sqrt (1-1/V^2) where V^2 = (dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2. I would interpret V^2 > 1 to be a superluminal velocity.