Einstein Vacuum Equation, Vacuum Constraint Equations

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.
  • #1
bookworm_vn
9
0
Having a Lorentzian 4-manifold, the Einstein vacuum equations of general relativity read

[tex]\overline R_{\alpha \beta} - \frac{1}{2}\overline g_{\alpha\beta}\overline R=0[/tex]​

where [tex]\overline R[/tex] the scalar curvature, [tex]\overline g_{\alpha\beta}[/tex] the metric tensor and [tex]\overline R_{\alpha\beta}[/tex] the Ricci tensor.

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

[tex]R - |k|^2 + ({\rm trace} \; k)^2=0[/tex]​

and

[tex]\nabla^\beta k_{\alpha\beta} - \nabla_\alpha {\rm trace} \; k=0[/tex]​

where [tex]R[/tex] is the scalar curvature of [tex]M[/tex], and [tex]k[/tex] 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 [tex]g[/tex] and [tex]k[/tex].

The point is how to derive these Vacuum Constraint Equations. Thank you very much.
 
Physics news on Phys.org
  • #2
Have you seen pages 258-259 of Wald?
 
  • #3
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 .
 

Related to Einstein Vacuum Equation, Vacuum Constraint Equations

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.

Similar threads

  • Special and General Relativity
Replies
32
Views
3K
  • Special and General Relativity
Replies
1
Views
808
  • Special and General Relativity
Replies
1
Views
1K
  • Special and General Relativity
Replies
7
Views
1K
  • Special and General Relativity
Replies
5
Views
1K
  • Special and General Relativity
Replies
8
Views
372
  • Special and General Relativity
2
Replies
59
Views
3K
  • Special and General Relativity
Replies
2
Views
1K
  • Special and General Relativity
Replies
5
Views
409
  • Special and General Relativity
2
Replies
62
Views
4K
Back
Top