1. What exactly is the cosmological constant, what is its value (or how do you derive it if it is something that must be derived for various situations) and how do I know when to use it and when not to use it in the Einstein field equations?(adsbygoogle = window.adsbygoogle || []).push({});

I ask this last question because sometimes I see the equations with just:

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

while other times I see them written as:

R_{[itex]\mu[/itex][itex]\nu[/itex]}- [itex]\frac{1}{2}[/itex]g_{[itex]\mu[/itex][itex]\nu[/itex]}R + g_{[itex]\mu[/itex][itex]\nu[/itex]}[itex]\Lambda[/itex] = [(8[itex]\pi[/itex]G)/c^{4}] T_{[itex]\mu[/itex][itex]\nu[/itex]}

(Sometimes I also see the equations with or without the c^{4}term.)

When do I use the cosmological constant, and when do I leave it out? What is the value of the constant, or how do I derive it? Why do people sometimes omit the speed of light term in the equations (though I'd guess that it is probably due to the c=1 convention, but correct me if I am wrong).

2. How do they get that c = 1 convention to make sense? What units are they using? I originally thought that they were saying that c = 1 light second per second, but I later found out that they were still using the SI unit meters.

How is that? c ≈ 3 * 10^{8}m/s. Do the units for c affect the units that you must use throughout the field equations? For the stress energy tensor for an electromagnetic field, I have seen multiple sources use CGS units as opposed to SI units. Does the c = 1 convention have any affect on the types of units that you must use for this tensor or is it your choice on whether you want to use SI or CGS?

3. In multiple sources that I have seen, people have used the (+ - - -) signature for tensors such as the stress energy momentum tensor. For example: In the electromagnetic stress energy momentum tensor, all of the elements that had a 0 as one of the indices were positive while all the purely spatial elements were negative.

Does this mean that if I want to use the (- + + +) signature that all of the elements that are purely spatial will be positive while all of the elements that have a 0 as one of the indices will be negative? I ask this because I am relatively new to deriving tensors in 4D (with time) and the only 4D tensor I ever saw before recently was the Minkowski metric tensor which is all 0 except for the diagonal. Therefore, I never actually got to see a full example of how the sign signature affects all of the elements including the ones that are not on the diagonal.

4. Finally, is it true that when solving the Einstein field equations, you are solving for the metric tensor? If so, how would I go about solving these, considering that all of the tensors on the whole space time curvature side of the equations are derived from the metric tensor? After all, if I don't have the metric tensor, then how would I derive the Cristoffel symbol which is needed to derive the Riemann tensor which is needed to derive the Ricci tensor which is needed to derive the curvature scalar?

Thank you

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# A few questions about Einstein's field equations

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