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From a problem the line element is:

$$ds^2 = -c^2e^{-2ax}dt^2 + dx^2+ dy^2+ dz^2$$

I found energy-momentum tensor ($T_{\mu\nu}$) from Einstein field equation by using the above line element. Only T$_{00}$ and T$_{11}$ survive, other $T_{\mu\nu}$'s are zero. I wonder that are these tensors that I found physically acceptable?

My answer:

$$T_{00}=c^2a^2e^{-2ax}c^4/(8\pi G)$$

$$T_{11}=a^2c^4/(16\pi G)$$

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  • $\begingroup$ Depends on how you define 'physically acceptable'. One way to constrain 'physically acceptable' energy-momentum tensors is to use energy conditions as mentioned in an answer below. $\endgroup$ – Avantgarde Apr 11 '19 at 12:16
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Exists some conditions called energy conditions. These conditions, roughly speaking, answer the question if some energy-momentum tensor are acceptable. Basically, there are some of then for massive particles (and for massless particles, but I will list some of for massive particles):

The time-like geodesic convergence condition: $$ R_{ab}V^{a}V^{b} \geq 0$$

The Weak Energy Condition WEC (for time-like vectors): $$ T_{ab}V^{a}V^{b} \geq 0$$

The Strong Energy Condition SEC (for time-like vectors):

$$ (T_{ab}-\frac{1}{2}Tg_{ab})V^{a}V^{b} \geq 0$$

You can check then if your tensor satisfies, principally, the WEC.

Also, some authors says that a physical solution of Einstein Field Equation must to satisfy both :

1) $G_{ab} = 8\pi T_{ab}$

2)$ T_{ab}V^{a}V^{b} \geq 0$

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The OP's metric is a space $AdS_2 \times E_2 $, a product of 2-dimensional anti-de Sitter spacetime and Euclidean plane. So, first, one of the components of energy-momentum tensor in the question has an erroneous factor of $2$ somewhere. And second, the energy density is constant everywhere and negative.

While the negative energy density may be considered unphysical, the $AdS_2$ space sometimes arises in GR as a limit of physically important solutions. For example, the near-horizon geometry of an extreme Reissner–Nordström black hole has the structure $AdS_2 \times S_2 $, a product of anti-de Sitter space and a sphere.

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