He most general form of the metric for a homogeneous, isotropic and st

[MarkovMarakov]

What is the most general form of the metric for a homogeneous, isotropic and static space-time?

For the first 2 criteria, the Robertson-Walker metric springs to mind. (I shall adopt the (-+++) signature)
$$ds^2=dt^2+a^2(t)g_{ij}(\vec x)dx^idx^j$$

Now the static condition. If I'm not mistaken, it means that the metric must be time-independent and invariant under time reversal $$t\to -t$$. So does that mean that the most general metric that satisfies all these 3 criteria is $$ds^2=dt^2+g_{ij}(\vec x)dx^idx^j$$ for some spatial metric $$g_{ij}(\vec x)$$?

Thank you.

 

[Mentz114]

What is the most general form of the metric for a homogeneous, isotropic and static space-time?

For the first 2 criteria, the Robertson-Walker metric springs to mind. (I shall adopt the (-+++) signature)
$$ds^2=dt^2+a^2(t)g_{ij}(\vec x)dx^idx^j$$

Now the static condition. If I'm not mistaken, it means that the metric must be time-independent and invariant under time reversal $$t\to -t$$. So does that mean that the most general metric that satisfies all these 3 criteria is $$ds^2=dt^2+g_{ij}(\vec x)dx^idx^j$$ for some spatial metric $$g_{ij}(\vec x)$$?

Thank you.

Surely all the spatial metric coefficients must be equal ?

 

[MarkovMarakov]

Perhaps I should impose the condition of constant curvature...? Is that enough?

 

[Mentz114]

Perhaps I should impose the condition of constant curvature...? Is that enough?

Probably. If the definition of homogeneity requires the same curvature everywhere. I'm sure someone will give you a better answer soon. I don't have my books at hand so I can't look it up.

 

[PeterDonis]

If the definition of homogeneity requires the same curvature everywhere.

I believe this is correct. However, I also believe that a homogeneous, isotropic, static metric can only be a solution of the Einstein Field Equation with a nonzero cosmological constant (e.g., the Einstein static universe). With a zero cosmological constant, a homogeneous, isotropic metric can't be static; it must be either expanding or contracting.

(Einstein introduced the cosmological constant in order to allow a static solution for the universe, but later, when the expansion of the universe was discovered, he called this "the greatest blunder of my life" because he missed the opportunity to predict an expanding universe before it was discovered.)

 

[WannabeNewton]

As Peter notes, a homogenous isotropic universe coming from the standard FLRW models cannot be static because the evolution equation $3\frac{\ddot{a}}{a}= -4\pi(\rho + 3P)$ tells us that for positive average mass density $\rho > 0$ and non-negative pressure $P\geq 0$, $\ddot{a} < 0$.

If we add a cosmological constant $\Lambda$ then according to problem 5.3 of Wald, static solutions exist iff $k = +1$ and $\Lambda > 0$. I haven't done the problem yet myself but you might find it instructive (there is also a second part investigating the instability of such universes which is where the real interest of the problem lies!).

EDIT: Just to add, for the $P = 0$ (dust) case, we can drop the isotropy and homogeneity and be more general with regards to the impossibility of a static solution. See here if you are interested: https://www.physicsforums.com/showthread.php?t=683601

 

[MarkovMarakov]

Thank you, everybody!