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Here, we have the field equations:

R

_{[itex]\mu[/itex][itex]\nu[/itex]}- [itex]\frac{1}{2}[/itex]g

_{[itex]\mu[/itex][itex]\nu[/itex]}R = ((8[itex]\pi[/itex]G)/c

^{4})T

_{[itex]\mu[/itex][itex]\nu[/itex]}

Now, notice that the curvature scalar R is apart of the equations. Now we know that:

R = g

^{[itex]\mu[/itex][itex]\nu[/itex]}R

_{[itex]\mu[/itex][itex]\nu[/itex]}

This means that we can express the equations as:

R

_{[itex]\mu[/itex][itex]\nu[/itex]}- [itex]\frac{1}{2}[/itex]g

_{[itex]\mu[/itex][itex]\nu[/itex]}g

^{[itex]\mu[/itex][itex]\nu[/itex]}R

_{[itex]\mu[/itex][itex]\nu[/itex]}= ((8[itex]\pi[/itex]G)/c

^{4})T

_{[itex]\mu[/itex][itex]\nu[/itex]}

Now, the g

_{[itex]\mu[/itex][itex]\nu[/itex]}and the g

^{[itex]\mu[/itex][itex]\nu[/itex]}cancel each other out.

This just leaves:

R

_{[itex]\mu[/itex][itex]\nu[/itex]}- [itex]\frac{1}{2}[/itex]R

_{[itex]\mu[/itex][itex]\nu[/itex]}= ((8[itex]\pi[/itex]G)/c

^{4})T

_{[itex]\mu[/itex][itex]\nu[/itex]}

This reduces to:

[itex]\frac{1}{2}[/itex]R

_{[itex]\mu[/itex][itex]\nu[/itex]}= ((8[itex]\pi[/itex]G)/c

^{4})T

_{[itex]\mu[/itex][itex]\nu[/itex]}

Finally, you should be able to then multiply both sides of the the equation by 2 to get:

R

_{[itex]\mu[/itex][itex]\nu[/itex]}= ((16[itex]\pi[/itex]G)/c

^{4})T

_{[itex]\mu[/itex][itex]\nu[/itex]}

Now, you have the Ricci tensor derived and then you can derive the metric tensor from here (thus solving the equations).

Would my logic in this process be accurate or is there some mathematical error within my logic that would render this invalid?