# Einstein Vacuum Equation, Vacuum Constraint Equations

• bookworm_vn
In summary, the conversation discusses the Einstein vacuum equations of general relativity and their relation to the Lorentzian 4-manifold. The equations involve the scalar curvature, metric tensor, and Ricci tensor. Additionally, there is a discussion on the Vacuum Constraint Equations which involve no time derivatives and serve as restrictions on the data. The conversation also mentions pages 258-259 of Wald, which provide further information on the derivation of these equations.
bookworm_vn
Having a Lorentzian 4-manifold, the Einstein vacuum equations of general relativity read

$$\overline R_{\alpha \beta} - \frac{1}{2}\overline g_{\alpha\beta}\overline R=0$$​

where $$\overline R$$ the scalar curvature, $$\overline g_{\alpha\beta}$$ the metric tensor and $$\overline R_{\alpha\beta}$$ the Ricci tensor.

By using the twice-contracted Gauss equation and the Codazzi equations of the Riemannian submanifold $$M$$, one finds that the normal-normal and normal-tangential components of the above Einstein vacuum equation are

$$R - |k|^2 + ({\rm trace} \; k)^2=0$$​

and

$$\nabla^\beta k_{\alpha\beta} - \nabla_\alpha {\rm trace} \; k=0$$​

where $$R$$ is the scalar curvature of $$M$$, and $$k$$ its second fundamental form. These equations, called the Vacuum Constraint Equations involve no time derivatives and hence are to be considered as restrictions on the data $$g$$ and $$k$$.

The point is how to derive these Vacuum Constraint Equations. Thank you very much.

Have you seen pages 258-259 of Wald?

hamster143 said:
Have you seen pages 258-259 of Wald?

Got it, thanks in advance, I am working in pure Maths so that I had not known the book due to Wald .

## 1. What is the Einstein Vacuum Equation?

The Einstein Vacuum Equation, also known as the Einstein Field Equation, is a set of equations in the theory of general relativity that describes the relationship between the curvature of space-time and the distribution of matter and energy within it.

## 2. What are the Vacuum Constraint Equations?

The Vacuum Constraint Equations, also known as the Bianchi identities, are a set of differential equations that arise from the symmetry properties of the Einstein Vacuum Equation. They are used to ensure the consistency and completeness of the theory of general relativity.

## 3. What is the significance of the Einstein Vacuum Equation?

The Einstein Vacuum Equation is significant because it provides a mathematical framework for understanding the behavior of gravity and the structure of space-time. It has been extensively tested and confirmed through various experiments and observations.

## 4. How are the Vacuum Constraint Equations derived?

The Vacuum Constraint Equations are derived from the Einstein Vacuum Equation by taking the divergence of the field equations. This process results in a set of equations that describe the conservation of energy and momentum in space-time.

## 5. What is the role of the Vacuum Constraint Equations in general relativity?

The Vacuum Constraint Equations play a crucial role in general relativity as they ensure that the theory is consistent and satisfies the laws of conservation of energy and momentum. They also provide a way to check the accuracy and validity of solutions to the Einstein Vacuum Equation.

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