Question about reverse tracing the Einstein field equations

[space-time]

From what I know, to get the reverse trace form of the Einstein field equations, you must multiply both sides by g^ab (I didn't have a lot of time to make this thread so I did not spend time finding the Greek letters in the latex).

This turns:

R_ab- $\frac{1}{2}$g_abR= kT_ab (where k= (8$\pi$G)/c⁴)

into this:

R- $\frac{1}{2}$R = kT

which equals:

$\frac{1}{2}$R = kT

which yields

R= 2kT= LT (I set L= 2k).

However, many sources say that the reverse trace is:

R= -LT

Why is it negative? Is it based on sign signature?

Note that I general, I use (- + + +) signature.

 

[Nugatory]

Your problem is that $g^{ab}g_{ab}=4$ when you do the summation.

 

[space-time]

Your problem is that $g^{ab}g_{ab}=4$ when you do the summation.

So then this yields:

R-2R= kT

which equals

-R = kT

which yields:

R = -kT

I presume that this is the correct process. Is this what you are getting at? If so then I get it now. Thank you.

 

[pervect]

So then this yields:

R-2R= kT

which equals

-R = kT

which yields:

R = -kT

I presume that this is the correct process. Is this what you are getting at? If so then I get it now. Thank you.

It looks right. I think Baez has some lecture notes that do the derivation online that go through this, if you haven't seen them already (I suspect it could be your source).

 

[sardar]

The sign of the reverse trace of the Einstein field equations is indeed based on the signature of the metric tensor, which determines the sign convention for the curvature tensor. In your example, you have used a (+ - - -) signature, which is commonly used in general relativity. In this case, the reverse trace form of the equations would indeed be R= -LT.

However, some sources may use a different signature convention, such as (- + + +) or (- - - -). In these cases, the sign of the reverse trace would be different. It is important to be aware of the signature convention being used in order to correctly interpret the equations.

Additionally, it is worth noting that the reverse trace form of the Einstein field equations is not always used in general relativity. It is often more convenient to work with the original form of the equations, and the reverse trace form is only useful in certain specific cases. So it is not necessarily always relevant to consider the sign of the reverse trace.

In summary, the sign of the reverse trace of the Einstein field equations is determined by the signature convention being used, and it is important to be aware of this convention in order to correctly interpret the equations.