# Physically acceptable energy-momentum tensor

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?

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

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

• 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. – Avantgarde Apr 11 '19 at 12:16

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

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.