Table of Connection Coefficients

[glamotte7]

Hello,

I'm trying to find a table of basic connection coefficients. My GREAT desire is to find a table that includes the metric, connection coeffs, Riemann tensor, Ricci tensor and Ricci scalar for NORDSTROM's 1913 scalar theory of gravity.

Does anyone know an online source of this info?

glamotte7

 

[Matterwave]

Since Nordstrom's theory is a scalar theory, wouldn't it lack these tensors? That's just my thought, but I don't know much about his theory.

 

[Ben Niehoff]

Nordstrom's theory is still a geometric theory of gravity. The metric is always a conformal factor times Minkowski space:

$$ds^2 = e^{2 \phi} \, (-dt^2 + dx^2 + dy^2 + dz^2)$$

It is very easy to compute the Christoffel symbols and curvature tensors for this metric. Why don't you try it!

 

[glamotte7]

For small ∅ this would be 1 + (2∅), correct?

What does the word conformal factor mean? Why is it advantageous to use an exponential as opposed to some other function? I've seen other important GR equations solved beginning with an arbitrary exponential. Why is this the best approach for a solution?

Lastly, I'm new here. How do you type math expressions like dx^2, etc?

 

[glamotte7]

Also, in Nordstrom's scalar theory, can't the scalar conformal factor be multiplied in an unequal fashion times -dt,dx,dy and dz? For example, can't dy and dz remain multiplied by 1 rather than multiplied by the factor, which in this case the factor would only affect dt and dx?

 

[Ben Niehoff]

Why is it advantageous to use an exponential as opposed to some other function?

The exponential is just to remind us that it has to be a positive number. Set $\phi = \log u$ if you want, it doesn't matter.

Also, in Nordstrom's scalar theory, can't the scalar conformal factor be multiplied in an unequal fashion times -dt,dx,dy and dz?

No.

 

[glamotte7]

Are you aware of a basic review article or reference on Nordstrom's scalar theory, accessible online?