Is This Spacetime Geometry Mathematically Conceivable?

In summary, the conversation revolves around the possibility of inventing 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 and other participants discuss the algebraic manipulation of the equation and its physical meaning, particularly in terms of proper time and velocity. The discussion concludes without a clear resolution on the feasibility of such a geometry.
  • #1
Eugene Shubert
22
0
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
 
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  • #2
isn't that just
ds2 = dt2- (dx2+dy2+dz2)
?
 
  • #4
Umm, I would agree with schwarzchildradius here. Just multiply through and suddenly you get rid of the nasty fractions.
 
  • #5
Yes I do remember elementary algebra, good for me. you can invert that equation.
 
  • #6
Would you like to expain in more detail how you think you can invert that fraction to get the required result?
 
  • #7
I thought flipping a fraction such as 1/3^2 would result in 3^-2.

Doesn't it change the exponent?
 
  • #8
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.
 
Last edited by a moderator:
  • #9
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.
 
  • #10
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.
 

1. What is spacetime geometry?

Spacetime geometry is a mathematical framework that describes the structure of the universe. It combines the concepts of space and time into a single entity, where space and time are no longer considered separate, but rather interconnected dimensions.

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Spacetime geometry is a crucial component of Einstein's theory of general relativity, which describes the gravitational force as the curvature of spacetime caused by massive objects. In this theory, the presence of matter and energy affects the shape of spacetime, and the motion of objects is determined by this curved geometry.

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While the mathematical concepts and equations of spacetime geometry can be difficult to visualize, there are various diagrams and illustrations that can help in understanding the concept. These visual aids often represent the curvature of spacetime as a rubber sheet, with massive objects creating dips and curves in the fabric of spacetime.

5. How does spacetime geometry impact our daily lives?

Spacetime geometry has significant implications in our daily lives, particularly in the fields of technology and communication. The principles of spacetime geometry are utilized in the development of GPS systems, which rely on precise measurements of time and space to determine location. Additionally, it also plays a role in our understanding of the universe and how it functions.

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