• Support PF! Buy your school textbooks, materials and every day products via PF Here!

Einstein field eqns, with cosmological const, Newtonian limit

  • Thread starter Mmmm
  • Start date
63
0
1. 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)

2. 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 guage 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]



3. 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:

Want to reply to this thread?

"Einstein field eqns, with cosmological const, Newtonian limit" You must log in or register to reply here.

Related Threads for: Einstein field eqns, with cosmological const, Newtonian limit

Replies
1
Views
776
Replies
5
Views
1K
  • Posted
Replies
0
Views
629
Replies
4
Views
16K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top