Einstein tensor fully written out

[greypilgrim]

Hi,

Does somebody know a link where the Einstein tensor is fully written out, i.e. only containing the metric and its derivatives? I'm just wondering how much is actually hidden in the notation.

 

[dextercioby]

There;s only message anyone can give: write it out yourself.

 

[Mentz114]

Yes, it takes ages. This is the Riemann tensor in terms of Christoffel symbols

${R^{r}}_{msq}=\Gamma ^{r}_{mq,s}-\Gamma ^{r}_{ms,q}+\Gamma ^{r}_{ns}\Gamma ^{n}_{mq}-\Gamma ^{r}_{nq}\Gamma ^{n}_{ms}$

the ',' before an index is differentiation. Contract on idexes r and s and you have the Ricci tensor, etc.

 

[bapowell]

http://en.wikipedia.org/wiki/Einstein_tensor

 

[sardar]

The Einstein tensor is a mathematical object used in Einstein's theory of general relativity to describe the curvature of spacetime. It is defined as:

Gμν = Rμν - 1/2 gμν R

where Rμν is the Ricci tensor, gμν is the metric tensor, and R is the scalar curvature. The Ricci tensor and scalar curvature are themselves defined in terms of the Christoffel symbols, which are in turn defined in terms of the derivatives of the metric tensor.

In order to fully write out the Einstein tensor, one would need to explicitly write out all of these components and their derivatives. This can be a very lengthy and complex calculation, as it involves multiple dimensions and a large number of terms.

However, it is important to note that the notation used in general relativity is not just a shorthand for these calculations, but also has deep physical significance. The Einstein tensor and its components represent the gravitational field and its effects on the curvature of spacetime. So while it may seem like a lot is hidden in the notation, it is actually a concise and powerful way of expressing complex physical concepts.

I hope this helps to answer your question. If you would like more specific information or resources on the full calculation of the Einstein tensor, I suggest consulting a textbook or research paper on general relativity.