# Reverse-tracing Einstein's field equation

• TheMan112
In summary, the equation for trace of a metric is 4 and for Ricci tensor it's R. The absolute value of |g| is 1 only in special relativity, in general relativity it's -1.
TheMan112
How do I rewrite Einsteins famous field equation

$$R_{ab} - \frac{1}{2} g_{ab} R = \frac{8 \pi G}{c^4} T_{ab}$$

into:

$$R_{ab} = \frac{8 \pi G}{c^4} (T_{ab} - \frac{1}{2} g_{ab} T)$$

I've tried experimenting with reverse-tracing the Ricci-scalar, but I just don't get the right equation. The trace from these equations would yield $$R=-T$$. Is this really correct?

Multiply both sides by $$g^{ab}$$ to get

$$\frac{1}{2}R = \kappa T$$

then substitute back.

Mentz114 said:
Multiply both sides by $$g^{ab}$$ to get

$$\frac{1}{2}R = \kappa T$$

then substitute back.

Ok, I insert the expression and get:

$$R_{ab} - \frac{8 \pi G}{c^4} g_{ab} T = \frac{8 \pi G}{c^4} T_{ab}$$

$$R_{ab} = \frac{8 \pi G}{c^4} \left(T_{ab} + g_{ab} T \right)$$

Which as you can see is not entirely right. Have I missed something very obvious?

Take trace of both sides of Einstein eq. Trace of the metric is 4. Trace of Ricci tensor is R. You get

R = - k T

then substitute that back for R and take the metric term on the other side.

I made a mistake, $$g^{ab}g_{ab} = -1$$.

TheMan112 said:
...
$$R_{ab} - \frac{1}{2} g_{ab} R = \frac{8 \pi G}{c^4} T_{ab}$$

Mentz114 said:
I made a mistake, $$g^{ab}g_{ab} = -1$$.

Okay, that fixes it.

$$g^{ab}g_{ab} = \delta^a_a = 4$$

smallphi said:
$$g^{ab}g_{ab} = \delta^a_a = 4$$

Yes, you're correct. It's |g| = -1.

Last edited:
Mentz114 said:
Yes, you're correct. It's |g| = -1.

How can |g| = abs(g) = -1 ? An absolute value can't be negative. For example a radius cannot be negative.

|g| = 1 only in Special Relativity. In GR, the value of |g| changes with the coordinate system used but it's signature (the signs of the eigenvalues when you diagonalize it) doesn't change, depending on the sign convention it's either (-, +, +, +) or (+, -, -, -).

Got it right now, thanks everybody.

## 1. What is "Reverse-tracing Einstein's field equation?"

"Reverse-tracing Einstein's field equation" refers to the process of using Einstein's field equations, which describe the relationship between the curvature of space-time and the distribution of matter and energy, to determine the distribution of matter and energy based on the observed curvature of space-time.

## 2. Why is "Reverse-tracing Einstein's field equation" important?

"Reverse-tracing Einstein's field equation" is important because it allows scientists to make predictions about the distribution of matter and energy in the universe based on the observed curvature of space-time. This can provide valuable insights into the structure and evolution of the universe.

## 3. How is "Reverse-tracing Einstein's field equation" used in scientific research?

"Reverse-tracing Einstein's field equation" is used in scientific research by inputting data on the observed curvature of space-time into the equations and solving for the distribution of matter and energy. This can help scientists understand the formation of galaxies, the expansion of the universe, and other important cosmological phenomena.

## 4. What challenges are involved in "Reverse-tracing Einstein's field equation?"

There are several challenges involved in "Reverse-tracing Einstein's field equation." One major challenge is obtaining accurate and precise measurements of the curvature of space-time, as this can be difficult to do with current technology. Additionally, the equations themselves are highly complex and require advanced mathematical and computational techniques to solve.

## 5. How does "Reverse-tracing Einstein's field equation" relate to the theory of general relativity?

"Reverse-tracing Einstein's field equation" is an important aspect of the theory of general relativity, as it allows scientists to apply the principles of general relativity to real-world observations and make meaningful predictions about the universe. It also helps to validate and refine the theory of general relativity by comparing its predictions to observations and data.

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