# Only vacuum is possible in the large $D$ limit of General Relativity?

+ 2 like - 0 dislike
98 views

The Einstein equations with a cosmological constant $\Lambda$ read as:

$R_{{\mu}{\nu}}-\dfrac{1}{2}Rg_{{\mu}{\nu}} + \Lambda g_{{\mu}{\nu}} =8\pi T_{{\mu}{\nu}}$

Therefore,

$R-\dfrac{D}{2}R+D\Lambda=8\pi T$

($D$ is the number of spacetime dimensions.)

Or,

$\Lambda = \dfrac{8\pi T}{D} + \bigg(\dfrac{D-2}{2D}\bigg)R$

Therefore,

$\Lambda = \bigg(\dfrac{D-2}{2D}\bigg)R_0$
where $R_0 := R_{vacuum}$

From the field equations,
$R_{{\mu}{\nu}}-\dfrac{1}{2}Rg_{{\mu}{\nu}} + \bigg(\dfrac{D-2}{2D}\bigg)R_0 g_{{\mu}{\nu}} =8\pi T_{{\mu}{\nu}}$

Therefore,
$8\pi T = \dfrac{D-2}{2} (R_0 - R)$

Now, in the large $D$ limit, the only way $T$ can be saved from diverging is to make $R$ approach $R_0$. This means that in the large $D$ limit, $R=R_{0}$ everywhere, i.e., the large $D$ limit of General Relativity admits only vacuum solutions.

I am posting this question to confirm if the conclusion I have reached is appropriate because I haven't come across any such claim elsewhere. Also, if this is appropriate then does it denote something interesting or more profound? (Put in other words, can it be associated with some rather known facts or principles?)

PS: I just noticed that in 2-dimensional case (1+1 dimensional case) also, only the vacuum solutions can exist and the cosmological constant also must vanish. (One can verify trivially by putting $D=2$ in the formula for $\Lambda$ and $T$ above.)

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOverfl$\varnothing$wThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.