Is this equation I made correct?

[zeromodz]

I wanted to make an equation that shows gravitational length contractions by using coordinates instead of a final length.

L^2 = X^2 + Y^2 + Z^2
L = √(X^2 + Y^2 + Z^2)


L = Lo * √(1 - 2GM / RC^2) (Gravitational length contraction)
L = √(X^2 + Y^2 + Z^2) * √(1 - 2GM / RC^2)
L = √(X^2 + Y^2 + Z^2 - (2GMX^2)/XC^2 - (2GMY^2/YC^2) - (2GMZ^2/ZC^2))
L = √(X^2 + Y^2 + Z^2 - (2GMX/C^2) - (2GMY/C^2) - (2GMZ/C^2))


L = √(X^2 + Y^2 + Z^2 -2GM(X + Y + Z) / C^2) <--------Final equation.

What do you think?

 

[Chronos]

The fatal flaw is you have assumed a background coordinate system.

 

[zeromodz]

The fatal flaw is you have assumed a background coordinate system.

Could you please elaborate? Why can't I use a background coordinate system?

 

[Chalnoth]

Could you please elaborate? Why can't I use a background coordinate system?

In General Relativity, coordinate systems are only valid locally. Curvature makes it so that you simply can't map any single coordinate system onto a whole space-time without running into problems.

To see why curvature does this, you can take the Earth as an example. If you start doing your calculations using the normal longitude/latitude coordinates we use on the surface of the Earth, you'll find that your calculations go haywire at the poles (because at the poles, every longitude maps onto just one point).

So we simply cannot take coordinates as being fundamental. Rather, coordinates are simply numbers we write down to describe some local region of space-time.

Finally, there's the issue that gravitational length contraction is an observer effect. Different observers will, in principle, see very different lengths. But a coordinate system isn't necessarily describing what anyone observer sees.

 

[Chronos]

For a detailed explanation, see Sten Odenwald's discussion at http://einstein.stanford.edu/content/relativity/q2442.html.