Why vacuum Einstein equations are hyperbolic equations?

In summary, the conversation discusses the hyperbolic nature of vacuum Einstein equations and its application in the Cauchy problem for General Relativity. It is explained that a hyperbolic partial differential equation is one that has a well-posed initial value problem. The scalar theory of gravitation, known as Nordstrom's gravitation, is mentioned as an example of a hyperbolic equation. The concept of a Cauchy problem is further discussed and it is suggested to refer to Witten's book for a full study. Another approach is presented using the Lorentzian metric to show the hyperbolic nature of the vacuum Einstein equations. However, it is mentioned that further study is needed to fully understand the hyperbolicity of
  • #1
bookworm_vn
9
0
Hello friends,

I am now interested in Einstein scalar field equations with a very little knowledge about Physics. I would like to ask why vacuum Einstein equations are hyperbolic equations? As far as I know that for [tex]3+1[/tex]-dimensional manifold [tex]V[/tex], we can convert the vacuum Einstein equations as a system of time derivative w.r.t. the induced metric and the second fundamental form of a [tex]3[/tex]-dimensional manifold, called hypersurface [tex]\Sigma[/tex]. What happen to non-vacuum case since we have an extra term, called stress-energy momentum tensor. Can you explain in details?

Thank you.
 
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  • #2
I learn from a book saying that the vacuum Einstein equation can be rewritten as

[tex]{\partial _t}{g_{\alpha \beta }} = - 2N{k_{\alpha \beta }} + {\mathbb{L}_X}{g_{\alpha \beta }}[/tex]

and

[tex]{\partial _t}{k_{\alpha \beta }} = - {\nabla _\alpha }{\nabla _\beta }N + N\left( {{R_{\alpha \beta }} + trk{k_{\alpha \beta }} - 2{k_{\alpha \gamma }}k_\beta ^\gamma } \right) + {\mathbb{L}_X}{k_{\alpha \beta }}[/tex]

however I am not sure how to make sure that the above system is hyperbolic. Thanks.
 
  • #3
bookworm_vn said:
Hello friends,

I am now interested in Einstein scalar field equations with a very little knowledge about Physics. I would like to ask why vacuum Einstein equations are hyperbolic equations? As far as I know that for [tex]3+1[/tex]-dimensional manifold [tex]V[/tex], we can convert the vacuum Einstein equations as a system of time derivative w.r.t. the induced metric and the second fundamental form of a [tex]3[/tex]-dimensional manifold, called hypersurface [tex]\Sigma[/tex]. What happen to non-vacuum case since we have an extra term, called stress-energy momentum tensor. Can you explain in details?

Thank you.

Actually a hyperbolic partial differential equation is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem, so the Cauchy problem for GR can simply answer your question. But historically, it is ineresting to know that the first scalar theory of gravitation, known as the 1th theory of Nordstrom's gravitation, had the field equation [tex]\square \phi = 0[/tex] with [tex]\square[/tex] being the D'Alembert's operator. As you can see, this is exactly the equation of a wave in vacuum that is hyperbolic because if on the line t=0, you have the first derivative of \phi w.r.t t and itself specified arbitrarily as the initial values, then there exists a solution for all time.

AB
 
  • #4
Altabeh said:
Actually a hyperbolic partial differential equation is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem, so the Cauchy problem for GR can simply answer your question. But historically, it is ineresting to know that the first scalar theory of gravitation, known as the 1th theory of Nordstrom's gravitation, had the field equation [tex]\square \phi = 0[/tex] with [tex]\square[/tex] being the D'Alembert's operator. As you can see, this is exactly the equation of a wave in vacuum that is hyperbolic because if on the line t=0, you have the first derivative of \phi w.r.t t and itself specified arbitrarily as the initial values, then there exists a solution for all time.

AB

Thanks, what I am expecting is to show that studying Cauchy problems for Einstein equations make sense. My advisor suggests me to show that the vacuum Einstein equation is indeed a hyperbolic equation but I am still not able. Having this fact, it is reasonable to study Cauchy problem for the Einstein equations.
 
  • #5
bookworm_vn said:
Thanks, what I am expecting is to show that studying Cauchy problems for Einstein equations make sense. My advisor suggests me to show that the vacuum Einstein equation is indeed a hyperbolic equation but I am still not able. Having this fact, it is reasonable to study Cauchy problem for the Einstein equations.

Yeah it is! In the scalar theory of Gravitation, the scalar curvature R vanishes in vacuum, i.e. R=0. So introducing this into the Einstein and Fokker [*] definition of the Ricci scalar,

[tex]-6(\square \phi)/\phi^3 = R[/tex] results in the wave equation again which is hyperbolic as discussed earlier. Note that here the vacu solution can be obtained from their field equation,

[tex]R=24\pi GT[/tex]

where you put T=0. (T, here, is the trace of the energy-momentum tensor and G is Newton's gravitational constant).

I think now it is clear why the vacu scalar field equation would be hyperbolic.

But in the tensor theory of Gravitation or GR, this is a little bit hard to explain as you are required to take advanced lessons for the Cauchy problem of GR which elegantly shows what is behind the name 'hyperbolic' for field equations not only in vacuum, but anywhere else.

For a full study of this problem, take a look at:

Witten L. (ed.) Gravitation: an introduction to current research (Wiley, 1962), chapter 4.

[*] A. Einstein and A.D. Fokker, Ann. d. Phys. 44, 321 (1914).
AB
 
  • #6
Altabeh said:
...

For a full study of this problem, take a look at:

Witten L. (ed.) Gravitation: an introduction to current research (Wiley, 1962), chapter 4.

[*] A. Einstein and A.D. Fokker, Ann. d. Phys. 44, 321 (1914).
AB

Oh thanks, I am reading. BTW, I have another approach: Consider the Lorentzian metric
[tex]d{s^2} = - dt \otimes dt + {g_{ij}}(x,t)d{x^i} \otimes d{x^j}[/tex].​
Einstein equations in vacuum, i.e., [tex]G_{ij} = 0[/tex] become
[tex]\frac{{{\partial ^2}{g_{ij}}}}{{\partial {t^2}}} + 2{R_{ij}} + \frac{1}{2}{g^{pq}}\frac{{\partial {g_{ij}}}}{{\partial t}}\frac{{{\partial _{pq}}}}{{\partial t}}{ - g^{pq}}\frac{{\partial {g_{ip}}}}{{\partial t}}\frac{{{\partial _{jq}}}}{{\partial t}} = 0.[/tex]​
It seems that the above eqn. is hyperbolic but I am not sure how to check. Thanks.
 
Last edited:

1. Why are vacuum Einstein equations considered hyperbolic equations?

The vacuum Einstein equations describe the behavior of spacetime in the absence of matter or energy. These equations are considered hyperbolic because they have a well-defined direction of time, meaning that information can only flow in one direction. This is a key characteristic of hyperbolic equations.

2. What are the advantages of vacuum Einstein equations being hyperbolic?

The hyperbolic nature of the vacuum Einstein equations allows for a unique solution to be obtained for a given set of initial conditions. This is important in predicting the behavior of spacetime in the absence of matter or energy, and is crucial for understanding phenomena such as black holes and gravitational waves.

3. Can you explain the difference between hyperbolic and parabolic equations?

Hyperbolic equations have a well-defined direction of time, as mentioned earlier. In contrast, parabolic equations do not have this characteristic and instead describe phenomena that are influenced by both past and future conditions. This is why hyperbolic equations are more suitable for modeling dynamic systems.

4. Are there any real-world applications of vacuum Einstein equations being hyperbolic?

Yes, the vacuum Einstein equations have been used to predict and explain various phenomena in our universe, such as the behavior of black holes, gravitational waves, and the expansion of the universe. These equations are also important in the study of general relativity and its applications in modern physics.

5. Is it possible for vacuum Einstein equations to not be hyperbolic?

Yes, in certain cases, the vacuum Einstein equations can become degenerate or elliptic rather than hyperbolic. This occurs when there is a lack of a well-defined direction of time, such as in stationary spacetimes. These cases require different mathematical techniques for solving the equations and have different implications for the behavior of spacetime.

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