Einstein field eqns, with cosmological const, Newtonian limit

In summary, the question asks to prove that in the Newtonian limit, the equation G_{\alpha \beta} + \Lambda g_{\alpha \beta} = 8 \pi T_{\alpha \beta} reduces to \nabla^2 \phi = 4\pi \rho + \Lambda. This is achieved by using the equations g^{\alpha \beta} = \eta_{\alpha \beta} + h_{\alpha \beta} and G_{\alpha \beta} = -\frac{1}{2}\nabla^2 \overline{h}_{\alpha \beta}, along with the Newtonian limits for T_{00} and \overline{h}_{
  • #1
Mmmm
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Homework Statement


This question is a slightly customised version of Q18(a) P212 of Schutz

Prove that
[tex]G_{\alpha \beta} + \Lambda g_{\alpha \beta} = 8 \pi T_{\alpha \beta}[/tex]
in the Newtonian limit reduces to
[tex]\nabla^2 \phi = 4\pi \rho + \Lambda[/tex]

(I found this result in another text, using it the remaining parts of the question in the book work nicely)

Homework Equations



for weak gravity
[tex]g^{\alpha \beta} = \eta_{\alpha \beta} + h_{\alpha \beta}[/tex]

where
[tex]\eta_{\sigma \alpha} h^\sigma _\beta = h_{\alpha \beta}[/tex]

using lorentz gauge for stationary T
[tex]G_{\alpha \beta} = -\frac{1}{2}\nabla^2 \overline{h} _{\alpha \beta}[/tex]

where
[tex]\overline{h}_{\alpha \beta} = h_{\alpha \beta}-\frac{1}{2} \eta_{\alpha \beta}{h^\lambda} _\lambda[/tex]

Newtonian limits
[tex]\left T_{00}\right > \left T_{0i}\right > \left T_{ij}\right[/tex]

[tex]\left \overline{h}_{00}\right > \left \overline{h}_{0i}\right > \left \overline{h}_{ij}\right [/tex]

[tex]T_{00} \approx \rho[/tex]
[tex]\overline{h}_{00} \approx -4\phi [/tex]
[tex]{h}_{00} \approx -2\phi [/tex]



The Attempt at a Solution



[tex]G_{\alpha \beta} + \Lambda g_{\alpha \beta} = 8 \pi T_{\alpha \beta}[/tex]

using the above:

[tex]\Rightarrow -\frac{1}{2}\nabla^2 \overline{h}_{\alpha \beta} + \Lambda (\eta_{\alpha \beta} + h_{\alpha \beta}) = 8 \pi T_{\alpha \beta}[/tex]

non trivial eqn whaen [itex]\alpha = \beta =0[/itex]

[tex]\Rightarrow -\frac{1}{2}\nabla^2 \overline{h}_{00} + \Lambda (\eta_{00} + h_{00}) = 8 \pi T_{00}[/tex]

Newtonian limit
[tex]\Rightarrow -\frac{1}{2}\nabla^2 (-4\phi) + \Lambda (-1 + -2\phi) = 8 \pi \rho[/tex]

[tex]\Rightarrow \nabla^2 (\phi) = 4 \pi \rho + \frac{1}{2}\Lambda +\Lambda \phi[/tex]

it should be
[tex]\nabla^2 \phi = 4\pi \rho + \Lambda[/tex]

as you can see something has gone a bit wrong somewhere.
if that [itex] h_{00}[/itex] were -1 it would work...
 
Last edited:
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  • #2
I think it maybe something to do with the fact I'm ignoring the other components of T_{\alpha \beta} and h_{\alpha \beta}Any help would be greatly appreciated.
 

1. What are the Einstein field equations?

The Einstein field equations are a set of ten equations in general relativity that describe the relationship between the curvature of spacetime and the distribution of matter and energy within it.

2. What is the role of cosmological constant in these equations?

The cosmological constant, represented by the Greek letter lambda (Λ), was originally introduced by Einstein as a term in the equations to counteract the effects of gravity and maintain a static universe. However, it is now understood to be a measure of the energy density of the vacuum and plays a crucial role in the expansion of the universe.

3. What is the Newtonian limit of the Einstein field equations?

The Newtonian limit refers to the approximation of the Einstein field equations in the case of weak gravitational fields and low velocities, where they reduce to the familiar equations of Newtonian gravity.

4. How do the Einstein field equations explain the curvature of spacetime?

The Einstein field equations explain the curvature of spacetime by relating it to the distribution of matter and energy through the curvature tensor. This curvature is what we experience as gravity.

5. What are some important implications of the Einstein field equations with cosmological constant?

The Einstein field equations with cosmological constant have several important implications, including the prediction of an expanding universe, the existence of black holes, and the possibility of a multiverse. They also provide the foundation for our understanding of gravity and the structure of the universe on a large scale.

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