Showing that the Einstein Tensor has zero divergence

R_{mn} \equiv 0Using the definition of the covariant derivative, we can write the first two terms as:g^{in}R^{k}_{ikn} + g^{km}R^{k}_{klm} = g^{in}R^{k}_{ik;n} + g^{km}R^{k}_{kl;m}Now, let's use the definition of the Ricci tensor again:R^{k}_{ik;n} = R_{in;n} and R^{
  • #1
kudoushinichi88
129
2

Homework Statement


We have

[tex]R_{iklm;n}+R_{iknl;m}+R_{ikmn;l} \equiv 0[/tex]

Show that by multiplying above with [itex]g^{im}g^{kn}[/itex]

we'll get

[tex]\left( R^{ik}-\frac{1}{2} g^{ik} R \right)_{;k}[/tex]


2. The attempt at a solution

[tex]g^{im}g^{kn} \left( R_{iklm;n}+R_{iknl;m}+R_{ikmn;l} \right) \equiv 0[/tex]

[tex]g^{im}R_{i} ^{n}_{lm;n}+g^{kn}R^{m}_{knl;m}+\frac{\partial R}{\partial x^l} \equiv 0 [/tex]

[tex]R^{n}_{l;n}+R^{m}_{l;m}+\frac{\partial R}{\partial x^l} \equiv 0[/tex]

then I'm stuck, not sure how to proceed. Honestly I'm not sure if my contractions are correct either. Please help?
 
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  • #2


Firstly, your contractions are correct so far. To proceed, let's expand out the first two terms:

R^{n}_{l;n} = g^{in}R_{iln} and R^{m}_{l;m} = g^{km}R_{klm}

Substituting these into our equation, we get:

g^{in}R_{iln} + g^{km}R_{klm} + \frac{\partial R}{\partial x^l} \equiv 0

Next, let's use the definition of the Ricci tensor:

R_{il} = R^{k}_{ikl}

Substituting this into our equation, we get:

g^{in}R^{k}_{ikn} + g^{km}R^{k}_{klm} + \frac{\partial R}{\partial x^l} \equiv 0

Now, let's use the definition of the Ricci scalar:

R = g^{il}R_{il}

Substituting this into our equation, we get:

g^{in}R^{k}_{ikn} + g^{km}R^{k}_{klm} + \frac{\partial}{\partial x^l} \left(g^{il}R_{il} \right) \equiv 0

Using the product rule for derivatives, we can expand the last term as:

g^{in}R^{k}_{ikn} + g^{km}R^{k}_{klm} + g^{il}\frac{\partial R_{il}}{\partial x^l} + R_{il} \frac{\partial g^{il}}{\partial x^l} \equiv 0

Now, let's use the definition of the Christoffel symbols:

\Gamma_{il}^{m} = \frac{1}{2} g^{mn} \left(\frac{\partial g_{il}}{\partial x^n} + \frac{\partial g_{nl}}{\partial x^i} - \frac{\partial g_{in}}{\partial x^l} \right)

Substituting this into our equation, we get:

g^{in}R^{k}_{ikn} + g^{km}R^{k}_{klm} + g^{il}\frac{\partial R_{il}}{\partial x^l} + \frac{1}{2} g^{mn} \left(\
 

1. What is the Einstein Tensor?

The Einstein Tensor is a mathematical object in the theory of general relativity that describes the curvature of spacetime. It is defined as a combination of the Ricci Tensor and the scalar curvature.

2. Why is it important to show that the Einstein Tensor has zero divergence?

In general relativity, the Einstein Tensor is used to describe the gravitational field and its interactions with matter. If the Einstein Tensor has a non-zero divergence, it would indicate the presence of a source or sink of gravitational energy, which would violate the principle of energy conservation. Therefore, showing that the Einstein Tensor has zero divergence is crucial in maintaining the consistency of the theory.

3. How is the divergence of the Einstein Tensor calculated?

The divergence of a tensor is calculated using the divergence operator, which is a mathematical operation that involves taking the partial derivatives of the components of the tensor. In the case of the Einstein Tensor, the divergence is calculated by taking the partial derivatives of the components of the Ricci Tensor and the scalar curvature, and combining them in a specific way.

4. What is the physical meaning of a zero divergence Einstein Tensor?

A zero divergence Einstein Tensor indicates that the gravitational field is in a state of equilibrium, with no sources or sinks of gravitational energy. This is consistent with the principle of energy conservation, as it means that the total energy of the gravitational field remains constant.

5. How is the zero divergence of the Einstein Tensor proved?

The zero divergence of the Einstein Tensor can be proved using differential geometry and the Einstein field equations. It involves showing that the partial derivatives of the components of the Ricci Tensor and the scalar curvature cancel out when combined, resulting in a zero divergence. This proof is essential in confirming the validity of the theory of general relativity.

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